Manchester City came from behind to draw 1-1 at home to Everton, but will feel like they should have won after missing two penalties.
City could have opened the scoring shortly before half-time when they were given a penalty after Phil Jagielka had tripped David Silva, but Maarten Stekelenburg was on hand to turn away Kevin De Bruyne's spot kick.
And City were made to pay for that miss in the second half when Romelu Lukaku ran from the halfway line before finishing calmly past Claudio Bravo.
The hosts had a chance to equalise when they were awarded their second spot kick of the day after Sergio Aguero had been brought down by Jagielka, only for Aguero to also see his effort turned away by Stekelenburg.
But City did finally find the back of the net in the 72nd minute when substitute Nolito beat the Everton keeper with a header as his side ultimately endured a third consecutive game without victory.
Arsenal moved level on points with City at the top of the Premier League table after defeating Swansea 3-2 at the Emirates Stadium in Bob Bradley's first game in charge of the Swans.
The Gunners broke the deadlock in the 26th minute through Theo Walcott, who bagged his fifth goal in five games for the club, poking home from close range after Swansea had failed to deal with the danger.
And Walcott soon made it six goals in five games a few minutes later, again capitalising on a mistake at the back before prodding in from just a few yards out.
Swansea pulled one back through Gylfi Sigurdsson, curling an effort from the edge of the area past Petr Cech, but Mesut Ozil restored Arsenal's two-goal advantage in the second half after volleying a cross into the roof of the net.
Borja Baston soon grabbed a second for the Swans, finshing off Modou Barrow's cross, but Arsenal were able to hold on for a sixth straight Premier League victory despite seeing Granit Xhaka dismissed for bringing down Barrow.
Diego Costa, Eden Hazard and Victor Moses were all on target for Chelsea as they brushed aside Leicester City with a 3-0 victory at Stamford Bridge.
The Blues raced into the lead inside just seven minutes as Nemanja Matic flicked on a corner at the near post, with Costa arriving unmarked in the middle to fire past Kasper Schmeichel for his seventh Premier League goal of the season.
Hazard made it two in the 33rd minute, taking advantage of some sloppy Leicester defending before rounding Schmeichel and slotting into an empty net.
And Moses rounded off the scoring in the 80th minute, firing past Schmeichel and into the back of the net after being fed in by a brilliant Nathaniel Chalobah pass.
Dele Alli scored a last-minute equaliser to secure Tottenham Hotspur a 1-1 draw against West Bromwich Albion at the Hawthorns.
Tottenham did everything but score in the first half, dominating the play with Dele Alli and Christian Eriksen both being denied by fine saves from Ben Foster.
And Spurs continued their search for a goal in the second half, with Moussa Sissoko and Eriksen both going close but unable to break the deadlock.
With just eight minutes left on the clock, Nacer Chadli came back to haunt his former club, blasting home from close range after Hugo Lloris had denied James McClean.
However, Alli had the final say, poking home from close range for Spurs after the ball had ran loose inside the area.
Joe Allen scored a brace as Stoke City secured their first Premier League win of the season with a 2-0 victory at home to fellow strugglers Sunderland to temporarily climb out of the relegation zone.
Allen needed just eight minutes before getting on the score sheet for Stoke, for the third successive Premier League game, arriving in the box to get on the end of Marko Arnautovic's delivery.
And the Wales international grabbed his second of the afternoon on the stroke of half-time, firing into the back of the net from the edge of the box after a corner had been cleared.
But West Ham's 1-0 win at Crystal Palace in the late game put Stoke back down to 18th in the table.
Manuel Lanzini neatly flicked home the only goal of the game after 19 minutes to give the Hammers a first league success in six outings.
Christian Benteke should have equalised from the penalty spot in the last minute of the first half but nonchalantly steered his spot-kick wide.
Palace could not capitalise on Aaron Cresswell's 75th-minute dismissal -- his two yellow cards coming within 51 seconds -- although they needed a superb save from Adrian in stoppage time to keep out a Connor Wickham effort and preserve the three points.
Mike Phelan's first game since being appointed permanent Hull City boss ended in defeat as Bournemouth ran out 6-1 winners at the Vitality Stadium.
The Cherries opened the scoring through Charlie Daniels, who was on hand to fire home the rebound after Junior Stanislas' free kick had come back off the post.
Hull drew level through Ryan Mason, with the midfielder's strike deflecting past Artur Boruc, however, Bournemouth were soon back in front thanks to Steve Cook, who headed home Stanislas' delivery.
Stanislas then got on the score sheet himself just minutes later, slotting into the roof of the net from the penalty spot, before finding the back of the net once more in the second half, with Callum Wilson and Dan Gosling adding a fifth and sixth late on.