Ten-man Chelsea held on for a 1-1 draw at Liverpool after Reece James' controversial dismissal, while Mikel Arteta urged his Arsenal flops to "look in the mirror" after a dismal 5-0 thrashing at Manchester City on Saturday.
Liverpool held by 10-man Chelsea, Man City pile on misery for Arsenal
Anthony Taylor shows a red card to Chelsea defender Reece James
European champions Chelsea took the lead through Kai Havertz's looping header from James's corner in the 22nd minute.
But a hard-fought clash of title contenders turned when James was sent off in first half stoppage-time.
The Chelsea defender, stationed on the goal-line, blocked Sadio Mane's effort with his thigh and then his arm after Joel Matip's header had hit the bar.
Referee Anthony Taylor was advised to check with the pitchside monitor and he awarded a penalty before sending off James.
Chelsea complained furiously but Mohamed Salah stepped up to convert the spot-kick before another angry protest from Blues defender Antonio Rudiger and goalkeeper Edouard Mendy sparked a penalty area melee.
Liverpool dominated the second half but Chelsea defended superbly, with Mendy making several fine saves.
Both sides remain unbeaten after seeing their 100 percent records come to an end in what was the first chapter of what should be a gripping title race.
"With pure will, we took the heat out of the game. That was team effort at its very best," Chelsea boss Thomas Tuchel said.
"The red card spoils the game. I have no solution for it because maybe it is a red card because he denies a chance but it is also a deflection. I would have liked a longer (VAR) check."
Liverpool manager Jurgen Klopp said: "Everything was enjoyable apart from the result. It was a great first half from us and actually a good second one because it's so tricky when you play against nine defenders.
"Could we have done better? Yes. But it's early in the season."
Arsenal have lost their first three league games and are yet to score a goal this term.
Arteta's side sit bottom of the league after their worst start to a season since 1954 as pressure mounts on their under-fire boss.
Arteta's switch to a back three failed to halt a slick City attack that suggested they are more than capable of defending their Premier League title this season despite missing out on Harry Kane and Cristiano Ronaldo.
Ilkay Gundogan opened the floodgates before Ferran Torres pounced on shocking Arsenal defending to double City's lead inside 12 minutes.
Arsenal's task was made even harder when Granit Xhaka needlessly lunged at Joao Cancelo and was shown a straight red card.
Gabriel Jesus made it 3-0 before half-time before Rodrigo and Torres, with his second of the afternoon, rounded off the scoring.
"I'm really disappointed with the things that happened on the pitch and the summary after three games, losing all of them, doesn't make it any easier," Arteta said.
"It is time to reflect and look in the mirror, each of us, and try to change the dynamic straight away because we need to pick up results."
Despite Arteta's woes, Manchester City boss Pep Guardiola insisted Arsenal should keep faith with his former assistant.
"I know, completely, his talent. I am pretty sure the moment will come when everything will be back and he will do a job because he is an excellent manager," Guardiola said.
Everton beat Brighton 2-0 to extend their unbeaten start under former Liverpool boss Rafael Benitez.
A controversial appointment to replace Carlo Ancelotti in the close-season due to his Anfield links, Benitez is winning over his new fans after goals from Demarai Gray and Dominic Calvert-Lewin on the south coast.
After failing to win his first two league games in charge of Crystal Palace, while also crashing out of the League Cup in midweek, Arsenal and France legend Patrick Vieira watched his team earn a 2-2 draw at in-form West Ham.
Promoted Brentford remain unbeaten after a 1-1 draw at Aston Villa.
Brentford's Ivan Toney got the opener and Emi Buendia scored his first Villa goal to equalise.
Southampton snatched a dramatic 2-2 draw at Newcastle thanks to James Ward-Prowse's stoppage-time equaliser.
Leicester won 2-1 at Norwich, with Jamie Vardy putting the Foxes ahead and Teemu Pukki equalising from the penalty spot before Marc Albrighton wrapped up the points.