The defending champions successfully chased down the modest target with skipper Gambhir leading the way with a composed knock of 42 off 29 balls with five boundaries and a six.
Kallis and Tiwary made 23 apiece.
Left arm spinner Nadeem claimed two for 22 while Nehra and Botha snared a wicket apiece.
Morgan hit the winning run as KKR won the first match of IPL6 by six wickets with eight balls to spare against Delhi Daredevils at Eden Gardens on Wednesday.
Yusuf Pathan remained unbeaten on 18 off 16 balls while Morgan was not out on 14 off 15 balls.
KOLKATA KNIGHT RIDERS INNINGS
Kolkata are racing to victory with Yusuf Pathan hitting Russell inside out for a six.
Can Daredevils pull off a miracle as Kolkata lose wickets in quick succession.
Kolkata are committing hara kiri as Tiwary departs attempting a paddle sweep off Nadeem which balloons in the air for Ojha to take an easy catch
Manoj Tiwary c Ojha b Nadeem 23
Tiwary made a run a ball 23 with two boundaries.
Morgan has opened his account with a boundary despite a desperate effort by Russell.
Botha strikes in his third over trapping Gambhir leg before.
Gautam Gambhir lbw b Botha 42
Gambhir faced 29 balls for his knock hitting five boundaries adding 41 runs for the third wicket.
Tiwary has hammered Russell through the covers for a boundary in the 13th over.
Five runs comes off the 11th over bowled by Botha as KKR are 80-2 after 11 overs.
Gambhir gets another boundary off Nadeem with a late cut to third man as Kolkata reach 75-2 after 10 overs.
Gambhir backs cuts Botha beautifully for another boundary in the ninth over as KKR are cruising in their run chase need 61 off 66 balls.
Tiwary opens his accounts with a boundary off left arm spinner Nadeem.
Kallis departs dragged a short delivery to Chand at midwicket off Nadeem as Daredevils gets a lucky break.
Jacques Kallis c Chand b Nadeem 23
Kallis faced 20 balls for his 23 which contained five boundaries.
Kallis has glanced Yadav to fine leg for another boundary as Pathan misfields.
Kallis has punched Nehra for another boundary through point in the final Powerplay over as Kolkata are on course making 46-1 after six overs.
Gambhir has cracked consecutive boundaries off Yadav in the fifth over clearing mid on and punching through the covers.
Kallis is taking on Nehra at the other end hitting a couple of boundaries in the fourth over, the first a sublime push past the bowler.
Kallis also gets into the act lofting Pathan for his first boundary.
Gambhir has taken on Pathan in the third over cracking him through point for a boundary and hitting the second six of the match over long on next ball.
Bisla hits the first boundary of the innings off the third delivery from Nehra but is beaten for pace next ball popping a catch to Chand.
Manvinder Bisla c Chand b Nehra 4
Irfan Pathan bowls a maiden over to Bisla as defending champions KKR begin their run chase in the IPL6 opener.
Nadeem is run out in the last ball of the innings as Delhi Daredevils are all out for 128 after being put into bat by Kolkata Knight Riders in the opening match of IPL6.
Jayawardene battled almost the entire innings to hold the Delhi innings together rocked by the first-ball dismissal of Chand.
The former Sri Lanka captain made 66 off 52 balls with eight boundaries and a six with opener Warner (21) being the only other batsman to enter into double figures.
Daredevils lost wickets at regular intervals with West Indian off spinner Sunil Narine claiming four for 13.
Bhatia claimed two for 23 while Lee's two wickets cost 40.
"The ball came in nicely. I was happy," said Lee who sent the off stump cartwheeling with the first delivery of IPL6.
DELHI DAREDEVILS INNINGS
Shabaz Nadeem run out 4
Narine gets his fourth wicket as Nehra is caught by Shukla for a duck.
Ashish Nehra c Shukla b Narine 0
Mahela's valiant knock comes to an end in the 19th over holing out to Narine in the deep off Lee.
Mahela Jayawardene c Narine b Lee 66
Mahela hit eight boundaries and a six facing 52 balls.
Mahela has hit the first six of IPL6 smashing Lee over midwicket in the 19th over followed by a boundary to long on.
Narine claims his third wicket taking a return catch to dismisss Russell.
Andre Russell c and b Narine 4
Mahela is cruising along paddling Lee's slower ball to square leg for the second boundary in the 17th over.
Mahela completes his half century off 46 balls with six boundaries.
Mahela cuts Lee past backward point for a boundary to bring the 100 of the Delhi innings in the 17th over.
Delhi skipper Mahela is on 44 off 41 balls with five boundaries as the second strategic time-out is taken with Daredevils on 97-6 after 15.2 overs.
Narine strikes on his return as Irfan Pathan holes out to Tiwary in the deep.
Irfan Pathan c Tiwary b Narine 4
Balaji also completes his fine spell returning figures of 4-0-20-1.
Bhatia completes a wonderful spell of 4-0-23-2.
Botha is adjudged leg before off Bhatia after attempting a reverse sweep.
Johan Botha lbw b Bhatia 7
Botha swings Shukla over the fine leg fielder for his first boundary in the 13th over.
Lee takes a tumbling catch in the deep off Bhatia to dismiss Ojha.
Naman Ojha c Lee b Bhatia 9
Mahela has dabbed Kallis past the vacant slip for a boundary in the 11th over.
Naman Ojha has steered Bhatia to third man for his first boundary.
It's a strategic time-out now with DD 59-3 after 8.5 overs.
IPL debutant Juneja hits his first boundary off Balaji but falls to a brilliant catch by Shukla at point next ball.
Manprit Juneja c Shukla b Balaji 8
Mahela has reached out to slice the first ball from Bhatia to third man for a boundary to bring up the 50 of their innings in the eighth over.
Balaji bowls a brilliant seventh over with new batsman Juneja in conceding just a couple.
Narine strikes with his fifth delivery as left hander Warner shuffles uneasily to edge to Kallis at lone slip.
David Warner c Kallis b Narine 21
Warner made 21 off 19 balls with four boundaries
Spin introduced for the first time as Narine comes into the attack and beats Mahela with his first delivery.
Mahela has greeted Balaji by glancing him to the fine leg fence in the fifth over with sublime timing.
Warner has flicked Lee with superb timing to square leg for a couple of boundaries in the fourth over.
Mahela cuts a short delivery from Lee over slips to third man for his second boundary in the third over.
Warner pulls Kallis to square leg for his second boundary.
It was not the first time a wicket has fallen in the first ball of the IPL, according to Mohandas Menon.
"Manoj Tiwary (KKR) was dismissed first ball from Chaminda Vaas (DC) in the opening match of IPL 2010 at MumbaiDYP" (@mohanstatsman)
The founder of IPL Lalit Modi has tweeted: "Looks like a good star to Ipl 6. Good luck to all teams. Looking forward to watching the games." (@LalitKModi)
Kallis into the attack now with the buzz at the ground already at fever pitch after the eventful first over.
Warner cracks the last ball of the first over bowled by Lee to the boundary through the covers.
Delhi skipper Jayawardene hits the first boundary of IPL6 clipping Lee to square leg.
U Chand b Lee 0
A sensational start to the sixth edition of the IPL at a raucuous Eden Gardens.
Brett Lee bowls Unmukt Chand off the first ball of IPL6.
Lee, Kallis, Morgan and Narine are Kolkata's four overseas players.
Delhi's four overseas players are Jayawardene, Warner, Andre Russell and Johan Botha.
Gambhir feels it is a different track from last year and hopefully get some early wickets.
Mahela is upbeat despite the string of injuries.
"It's a good opportunity for some of other guys. I was going to bat first. Looks a good wicket," said the former Sri Lanka captain who is keen to put some runs on the board returning from injury himself.
Johan Botha will make his first appearance for Delhi Daredevils along with Manpreet Juneja.
Delhi Daredevils XI: U Chand, D Warner, M Jayawardene, M Juneja, J Botha, N Ojha, A Russell, I Pathan, A Nehra, S Nadeem, U Yadav
Kolkata Knight Riders XI: G Gambhir, M Bisla, J Kallis, M Tiwary, Y Pathan, E Morgan, L Ratan Shukla, R Bhatia, B Lee, S Narine, L Balaji
Kolkata Knight Riders captain Gautam Gambhir has won the toss and electef to bowl first against Delhi Daredevils in the opening match of IPL6 on Wednesday.
After a glitzy Bollywood-style opening ceremony on Tuesday, the stage is set for the opening match of the sixth edition of the Indian Premier League (IPL6) at Eden Gardens between defending champions Kolkata Knight Riders and Delhi Daredevils.
Both teams have been hit by injury to key players or absence due to other engagements taking some of the gloss away from the match.
But once the action commence there is little doubt the entertainment on and off the field will be fever pitch rising to a crescendo in the 'City of Joy' nay around the world with millions of cricket fans likely to follow the action.
For the record, Daredevils will be without dashing opener Virender Sehwag who has been suffering back spasms and advised rest.
Kevin Pietersen will miss the entire tournament due to a knee injury sustained in New Zealand, while Jesse Ryder was set to fly to India before being assaulted in Christchurch.
Morne Morkel will also miss at least the first game because he is in South Africa representing Titans in the final stages of their domestic T20 competition.
KKR don't have a fully fit squad at their disposal either.
New Zealander Brendon McCullum is yet to join the squad in India, as he is recovering from a hamstring injury sustained during the England Tests.
James Pattinson, the Australia fast bowler, is resting at home, as advised by Cricket Australia following the Test tour of India.
Jacques Kallis is only just returning from injury and skipper Gautam Gambhir has recovered from jaundice.
Delhi's title dreams were shattered after losing against Chennai Super Kings in the semi-final last season.
The team is coached by Eric Simons alongside TA Sekar while Mahela Jayawarene leads the team on the field.
During the IPL Auction, Delhi Daredevils bought three overseas players including Jeevan Mendis, Jesse Ryder and Johan Botha.
Mahela Jayawardene (captain), Manprit Juneja, Varun Aaron, Ajit Agarkar, Gulam Bodi, Johan Botha, Unmukt Chand, Royston Dias, Muralidharen Gautam, Kedar Jadhav, Siddarth Kaul, Jeevan Mendis, Morne Morkel, Shahbaz Nadeem, Yogesh Nagar, Sujit Nayak, Pawan Negi, Ashish Nehra, Naman Ojha (wk), Irfan Pathan, Andre Russell, Virender Sehwag, Aristh Singhvi, Roelof van der Merwe, Venugopal Rao, David Warner, Umesh Yadav.
KOLKATA KNIGHT RIDERS
Led by one of the finest Indian batsmen Gautam Gambhir, the team in golden blue is coached by Australian veteran Trevor Bayliss alongside the bowling mentor Brett Lee.
While eyeing their second consecutive title this season, KKR bought two prolific players - Sri Lankan off-spinner Sachitra Senanayake and South African all-rounder Ryan McLaren - during the IPL 2013 auction.
Kolkata boasts two of the most consistent players in the batting and bowling department.
KKR skipper Gautam Gambhir was the second highest run scorer with 590 runs in 17 matches last season while mystery spinner Sunil Narine was the second highest wicket taker with 24 wickets in 15 matches last year.
Gautam Gambhir (captain), Brett Lee, Yusuf Pathan, Ryan McLaren, Lakshmipathi Balaji, Ryan Ten Doeschate, Manoj Tiwary, Sunil Narine, Shami Ahmed, Brendan McCullam (wk), Pradeep Sangwan, Jacques Kallis, Debabrata Das, Sachithra Senanayake, Shakib al Hasan, James Pattinson, Brad Haddin (wk), Eoin Morgan, Laxmi Ratan Shukla, Manvinder Bisla(wk), Iqbal Abdulla, Pradeep Sangwan, Rajat Bhatia, Sarabjeet Ladda.