A stunning late strike from Daniel Sturridge rescued a point for Liverpool in the 1-1 draw at Chelsea on Saturday while Manchester United's 3-1 defeat at West Ham United confirmed their worst top-flight start for 29 years and piled more pressure on manager Jose Mourinho.
The contrast with neighbours Manchester City could not be more marked, with Pep Guardiola's side beating Brighton & Hove Albion 2-0 to go ahead of Liverpool on goal difference.
Sturridge's late intervention at Stamford Bridge, which preserves his team's unbeaten start to the season, came moments after Juergen Klopp brought him on, perhaps in the hope he might replicate the goal he managed against the same opponents in the EFL Cup at Anfield on Wednesday.
Amazingly, Sturridge did exactly that, unleashing a 25-metre shot that found the top corner of the Chelsea net just when it looked like Liverpool, who had won all six of their previous league games this season, had run out of ideas.
"What a game, a spectacular football game," said Liverpool manager Klopp. "It was so deserved for Daniel (Sturridge), a really nice moment and I'm so happy for him."
It is the first time that Chelsea, for whom Eden Hazard scored with a low first-half drive, have not won after going ahead this season.
The result sets up nicely next weekend's Anfield clash with City -- whom they trail on goal difference. City brushed aside the limited challenge of Brighton with goals from Raheem Sterling and Sergio Aguero after the Argentine linked superbly with the Englishman to net his seventh goal in all competitions this season.
Liverpool and City are now nine points ahead of United, who are being quoted at an unlikely 150/1 to win the title this season.
United manager Jose Mourinho will be more concerned with where their next win will come from after another lacklustre showing ended with him lambasting his team for their "quality and mental approach" at West Ham.
The defeat marked another low after a turbulent week in which Paul Pogba was stripped of the vice-captaincy and United were dumped out of the EFL Cup by Derby.
Mourinho responded by fielding an unfamiliar 3-5-2 formation and omitting his misfiring striker Alexis Sanchez from the squad in favour of Anthony Martial.
But the result was an equally passionless performance that led former United captain Rio Ferdinand, who now works as a TV pundit, openly to question how long Mourinho will remain in charge.
"There are some big decisions to be made at United now," Ferdinand said on BT Sport. "There will be conversations at the top level, about the future of the manager and the squad, because the basics are not being done. There has to be something said."
West Ham took just five minutes to open the scoring with record signing Felipe Anderson producing an audacious back-foot flick.
An unlucky own goal increased their lead but even a superb strike from Marcus Rashford made little difference, with Marko Arnautovic scoring the winner.
Mourinho was predictably unhappy. "One goal is offside, the second goal is own goal and the third goal is referee's mistake but for the next game we have to have a better start," he said, adding that Pogba, who was substituted in the second half, was "trying hard". Elsewhere, goals by Jamie Vardy and Harry Maguire secured Leicester City's 2-0 win over Newcastle United, who have lost their opening four home games for the first time amid British media reports of a possible takeover fronted by Peter Kenyon, the former Manchester United and Chelsea chief executive.
Harry Kane scored twice, including a penalty, as Tottenham Hotspur inflicted further woe on bottom-placed Huddersfield Town. Arsenal also won 2-0, at home to Watford, although both goals came in the final 10 minutes as Unai Emery's side were made to work hard for their seventh successive win in all competitions.
The only worry was a hamstring injury to keeper Petr Cech, who was substituted and will be out for three weeks.
Wolverhampton Wanderers' Ivan Cavaleiro scored with his first touch in the Premier League as Southampton were beaten 2-0. The Portuguese capitalised on excellent work from Adama Traore as the promoted side continued their impressive start to the season.
Gylfi Sigurdsson had an eventful afternoon for Everton, atoning for his missed penalty with two goals in the 3-0 win over Fulham, who have never won at Goodison Park in the league.