Manchester United's poor start to the season continued as West Ham beat Jose Mourinho's men 3-1 at the London Stadium on Saturday.
An early Felipe Anderson strike and a Victor Lindelof own goal put the hosts 2-0 up before half-time.
Marcus Rashford pulled one back for the visitors against the run of play midway through the second half, before Marko Arnautovic strolled through the United defence three minutes later and slotted under David De Gea to seal the points.
West Ham are now unbeaten in four in all competitions and have found their rhythm under Manuel Pellegrini, but the pressure grows on Mourinho after United failed to win for the third time in a week.
Chelsea thought they had ended Liverpool's unbeaten start to the season, but Daniel Sturridge came off the bench to score a late equaliser in a 1-1 draw.
Eden Hazard gave Chelsea a first half lead, and remained firm for much of the match.
However, Sturridge scored a curling effort from 25 yards to ensure Liverpool left Stamford Bridge with a point.
While United continue to falter, Manchester City maintained their unbeaten start to the Premier League season with a 2-0 home win against Brighton.
Raheem Sterling was on hand to tap in Leroy Sane's low cross before the half-hour mark, and Sergio Aguero put the champions out of sight as he finished a wonderful team move after 65 minutes.
City held on to move to the top of the league.
Bernd Leno made his Premier League debut as Arsenal left it late to beat Watford 2-0.
The Germany international replaced the injured Petr Cech towards the end of the first half, and managed to deny the Hornets in an end-to-end encounter at the Emirates Stadium.
Alex Iwobi replaced Aaron Ramsey amid speculation over his Arsenal future, and his cross was turned in by Craig Cathcart to give Arsenal the lead late on.
Almost immediately from kick-off, Mesut Ozil sealed the win for the home side after more excellent play from Iwobi.
Tottenham beat Huddersfield 2-0 to continue their good form ahead of their Champions League clash with Barcelona next week.
Harry Kane made amends for missing a one-on-one in the first half by heading Spurs into the lead soon after from a Kieran Trippier cross, and quickly doubled the lead from the penalty spot after Danny Rose was brought down in the box.
Spurs made a double change at half-time with one eye on Barcelona, and rode a second half storm to move back into the top four.
Everton ended their five-match winless run in the league following a 3-0 victory at home to Fulham.
An uneventful first half was followed by a penalty at the start of the second, with Denis Odoi bringing down Dominic Calvert-Lewin, but Gylfi Sigurdsson's effort from the spot hit the bar -- the first time he has missed a penalty in the Premier League.
The Iceland midfielder atoned for his error, however, breaking the deadlock five minutes later, firing Everton into the lead from the edge of the 18-yard area.
Cenk Tosun came off the bench to end his 10-game stretch without a goal, heading home from Theo Walcott's deep cross.
Sigurdsson bagged his second of the game in stoppage time to give Marco Silva a much-needed win.
Newcastle failed to climb out of the relegation zone as they succumbed to a 2-0 home defeat to Leicester.
The visitors took the lead from the spot after Kenedy was adjudged to have handled in his own box, allowing Jamie Vardy to fire Leicester in front.
Harry Maguire compounded Newcastle's misery late in the second half after rising highest from James Maddison's corner.
Wolves continued their fine start back to life in the top flight with a 2-0 win at home to Southampton following last week's 1-1 draw at Manchester United
A largely uneventful game was settled with little over 10 minutes remaining, when substitute Ivan Cavaleiro was teed up by Raul Jimenez to fire home from the edge of the area.
Wolves doubled their lead through Jonny Otto to climb to eighth in the table.