Liverpool were sent spinning to a 2-0 defeat at promoted Burnley in the Premier League on Saturday while Chelsea came from behind to win 2-1 at Watford thanks to late goals from Michy Batshuayi and Diego Costa.
Elsewhere, Sergio Aguero and Nolito scored twice apiece as Manchester City won 4-1 at Stoke City to give Pep Guardiola's men their second straight victory while champions Leicester City were held to a 0-0 home draw by Arsenal.
Sam Vokes and Andre Gray scored first-half goals to put Burnley in command against Liverpool who were a pale shadow of the team that beat Arsenal 4-3 in their opening game of the season last weekend.
"How can I be happy with this?", said Liverpool manager Juergen Klopp. "Everybody needs to be more clinical.
"Burnley deserved to win with a very passionate performance. We have to say it wasn't enough today."
Vokes netted with a rasping shot in the second minute before Gray bagged his first top-flight goal with a crisp low strike from 18 metres.
Etienne Capoue gave Watford a 55th-minute lead against Chelsea after smashing a swerving shot into the top corner of the net.
Substitute Batshuayi equalised in the 80th minute with his first goal for the club before Costa, who scored a late winner against West Ham United on Monday, finished calmly three minutes from time after being found by another sub, Cesc Fabregas.
"I'm pleased to see a reaction from my players," said Chelsea manager Antonio Conte. "This win's very important for us as it gives everyone confidence."
Manchester City were arguably flattered by the 4-1 scoreline at Stoke.
Aguero, who missed two penalties in City's 5-0 Champions League playoff first-leg victory at Steaua Bucharest in midweek, opened the scoring from the spot after Ryan Shawcross impeded Nicolas Otamendi.
The Argentine then doubled his tally after 36 minutes, meeting a sumptuous Kevin De Bruyne free kick to send a fine header into the far corner of the goal.
Stoke fought back when Spanish forward Bojan beat keeper Willy Caballero from the spot but substitute Nolito struck twice late on for Manchester City.
"I am really impressed with the players as human beings and with their performance," said Guardiola.
Leicester were left fuming after they had two strong penalty appeals denied against Arsenal.
A staid first half came to life in the final minutes when Danny Drinkwater was felled by Laurent Koscielny but referee Mark Clattenburg waved away frantic appeals from Jamie Vardy and Riyad Mahrez, prompting a cacophony of boos from the home fans.
Substitute Ahmed Musa was then brought down by Hector Bellerin in the 88th minute as Leicester's counter-attacking pace put Arsenal under pressure, only for Clattenburg to again wave play on.
"I don't want to talk about the decisions," said City boss Claudio Ranieri. "The managers make mistakes, the referees make mistakes, the players make mistakes."
Victor Wanyama netted in the 83rd minute as Tottenham Hotspur beat Crystal Palace 1-0 while Everton won 2-1 at West Bromwich Albion.
Hull City became the first promoted side since Bolton Wanderers in 2001 to win their opening two matches when they triumphed 2-0 at Swansea City.