Former champions, Chelsea lost their 100 percent Premier League record when London rivals West Ham United held them to a 0-0 draw but at the Emirates stadium, fellow Lo don rival, Arsenal proved better with at 2-0 win over Everton FC.
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.
Despite having barely 30 percent of possession in the first half, West Ham finally came alive after 30 minutes with two good chances for Antonio.
Sent through by record signing Felipe Anderson he shot too high but then forced goalkeeper Kepa Arrizabalaga into a good save after being set up by Yarmolenko.
For all their dominance, Chelsea made barely one real opportunity before the interval when N’Golo Kante arrived to head wide from Willian’s smart volleyed cross.
In the second half it was two of Chelsea’s substitutes who went closest. Alvaro Morata’s first attempt hit goalkeeper Lukasz Fabianski in the face and later Ross Barkley’s curler was also kept out by the home goalkeeper.
Chelsea have another chance to knock Liverpool off the top when they play them at Stamford Bridge on Saturday — three days after the clubs meet in the League Cup at Anfield.
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.