Chelsea lost their 100 per cent Premier League record when London rivals West Ham United held them to a 0-0 draw on Sunday.
Needing a two-goal win to depose Liverpool at the top of the table, Chelsea struggled to make clear chances despite having the majority of possession and stayed third behind the Merseysiders and champions Manchester City.
The home side could have punished them but Michail Antonio, who was deputising for the injured Marko Arnautovic, missed two first-half chances and Andriy Yarmolenko headed wastefully wide.
West Ham, with only one win so far from their six matches, remain 17th, one place above the relegation zone.
Lacazette, Aubameyang help Arsenal edge past Everton
Arsenal broke down a determined Everton defence to win 2-0 on Sunday, with Alexandre Lacazette and Pierre-Emerick Aubameyang scoring two second-half goals in two minutes to give the Gunners their fourth straight Premier League win.
Everton had created most of the early chances as they looked for their first victory at Arsenal since January 1996, but they were unable to find their way past goalkeeper Petr Cech.
French striker Lacazette broke the deadlock just short of the hour mark for the Gunners, running in from the left before elegantly curling the ball into the far corner just beyond the reach of outstretched England keeper Jordan Pickford.
Gabon international Aubameyang doubled the lead two minutes later from what appeared to be an offside position, sending Pickford the wrong way from close range.
The win -- and Arsenal's first clean sheet of the season -- pushes the Gunners up to sixth in the table, while the Toffees stay in 12th place.