West Bromwich Albion won the early basement battle against Sunderland to heap more pressure on visiting manager Paolo Di Canio.
Newcastle missed out on the chance to go second, and subsequently dropped into the bottom half of the table, after losing to Hull City, while Aston Villa manager Paul Lambert enjoyed a happy return to Norwich City.
Early pacesetters Liverpool suffered their first defeat of the season after Dejan Lovren's second-half header earned Southampton a 1-0 win at Anfield.
Liverpool enjoyed the better of possession in the first half and came closest to scoring when captain Steven Gerrard's curling free-kick brought a brilliant one-handed save out of Artur Boruc.
Southampton stunned their hosts on 53 minutes when Lovren's header from a left-wing corner made it over the line after Gerrard's poor attempt to clear.
Reds keeper Simon Mignolet then produced several superb saves to prevent Southampton extending their lead as Mauricio Pochettino's side claimed a second win of the season.
Chelsea leapfrogged Liverpool on goal difference later in the day after second-half efforts from Oscar and John Obi Mikel clinched a 2-0 home win over west London neighbours Fulham at Stamford Bridge.
Darren Bent missed a golden opportunity for Fulham on 13 minutes when he shot straight at Petr Cech, while Chelsea could have had a penalty when Sascha Riether nudged Eden Hazard in the back.
The Blues, who had lost to Everton and Basel in the past week, took the lead on 52 minutes when Oscar scored from six yards out after David Stockdale had failed to hold Andre Schurrle's shot.
The home side appealed for another penalty when Ramires appeared to be caught by Fernando Amorebieta before they put the game to bed six minutes from time when Mikel volleyed home after John Terry had nodded down Frank Lampard's corner for his first Premier League goal for the Blues in his 185th appearance.
Romelu Lukaku marked his Everton debut with the winner after Leighton Baines' brace of magical free-kicks twice drew his team level in the 3-2 success at 10-man West Ham.
West Ham took the lead on 31 minutes when Matt Jarvis cut in from the left flank and teed up Ravel Morrison, whose 25-yard shot took a deflection and left Tim Howard wrong-footed for his first Premier League goal.
Everton drew level just past the hour mark when they were awarded a free-kick in a central position 30 yards from goal and Baines stepped up to curl a fine effort beyond Jussi Jaaskelainen.
The Hammers regained the lead 14 minutes from time when Mark Noble converted from the penalty spot after Everton substitute James McCarthy had tripped Kevin Nolan.
But five minutes later Noble was sent off for collecting a second yellow card for a trip on Ross Barkley and Baines, from a similar position to his first goal, stepped up and curled the free-kick in off the frame of the goal.
And with five minutes left Everton pinched the points when Kevin Mirallas' cross was headed in by Lukaku, who took a whack in the face for his troubles.
Stephane Sessegnon scored against his former club as West Brom secured a first win of the season after beating struggling Sunderland 3-0 at The Hawthorns.
Perhaps predictably, Sessegnon put West Brom in front on 20 minutes when he converted the rebound from around 10 yards out after Keiren Westwood parried Scott Sinclair's shot from Morgan Amalfitano's cross.
Sunderland suffered a major blow when, having used all their substitutes, striker Steven Fletcher was forced off through injury and soon after Westwood pushed the ball out and Liam Ridgewell was on hand to fire home in the 76th minute.
Having created two goals, Amalfitano got on the score-sheet himself in added time as he was played in on the right of the area and he lashed a low shot across Westwood.
Hull City twice came from behind before snatching a 3-2 victory over Newcastle at St James' Park to provide manager Steve Bruce with a satisfying trip back to the north east.
Newcastle were in front on 10 minutes when Moussa Sissoko ran with the ball from just inside his own half before crossing to the far post to Papiss Cisse, whose mis-hit volley was headed in by Loic Remy for his first goal since his loan move from Queens Park Rangers.
Hull were level on 26 minutes when the visitors worked the ball from right to left and Robbie Brady was found in space on the edge of the area to drill a low shot past Tim Krul.
But Newcastle were back in front a minute before half-time when Yohan Cabaye's shot was blocked by a Hull defender straight to Remy and he finished confidently first time beyond Allan McGregor.
Slack defending again allowed Hull to equalise for a second time three minutes after the break when no-one tracked Ahmed Elmohamady's run to the near post and his header from Brady's free-kick bounced in off the far upright.
The Tigers took the lead for the first time on 76 minutes when George Boyd pulled the ball back from the by-line and Aluko's excellent first-time volley flew in off the post.
Libor Kozak's first goal for Aston Villa secured a 1-0 win at Carrow Road after Norwich City winger Robert Snodgrass saw an early penalty saved by Brad Guzan.
In a fast start to the game, Andreas Weimann hit the post for Villa before Guzan kept out Snodgrass' spot-kick on five minutes after Ciaran Clark had been harshly penalised for handball.
The decisive goal arrived on the half-hour from Kozak, who scored at the second attempt after his initial shot had been parried by John Ruddy just moments after the Czech striker had replaced the injured Christian Benteke.