Kumble sent the Rajasthan Royals crashing to defeat when they started the defence of their crown at Newlands while Tendulkar led the Mumbai Indians to victory over the Chennai Super Kings.
Former India skipper Kumble took five for five as the Bangalore Royal Challengers gained a crushing 75-run win as the champions were bowled out for a paltry 58, the second lowest total in the short history of Twenty20 cricket.
"It was pretty disappointing," admitted Rajasthan captain Shane Warne. "As defending champions we expect better of ourselves. Our shot selection was poor."
Bangalore skipper Kevin Pietersen was overjoyed with the opening win.
"The boys played fantastic cricket," he said. "It was a great team effort and you can't buy Anil Kumble's experience in a shop."
Indian batting star Rahul Dravid made a fine 66 as Bangalore recovered from losing two wickets for no runs in the first three balls of their innings.
Bangalore made 133 for eight before the Royals were bowled out for 58, with no batsman making more than 11.
England all-rounder Dimitri Mascarenhas took 3-20 but Pietersen, who made 32, and Dravid paved the way for Bangalore's win. Dravid's 66 came off 48 balls with eight fours and a six.
Tendulkar hit an unbeaten 59 as he led the Mumbai Indians to a 19-run win against the Chennai Super Kings.
Tendulkar's 59 not out provided the foundation for Mumbai's 165 for seven after they were sent in on a blustery, grey day.
Former Australian opening batsman Matthew Hayden hit 44 and captain Mahendra Singh Dhoni 36 for Chennai but the 2008 runners-up could only manage 146 for seven.
Sri Lankan fast bowler Lasith Malinga bowled superbly to take three for 15 in four overs for Mumbai.