And, after five matches, Everton remains the only side unbeaten in the English Premier League.
Chelsea had been reeling from its worst start during the decade of Roman Abramovich's ownership, but second-half goals from Oscar and John Obi Mikel secured a 2-0 victory over Fulham.
The win put Chelsea on top ahead of Sunday's games, dislodging Liverpool, which was beaten 1-0 at Anfield by Southampton.
"I understand at this moment we are not playing very, very well, the results are not especially good," Chelsea manager Jose Mourinho said. "But the beauty of the Premier League is that ... everybody loses matches."
Not Everton — yet.
The Merseyside club staged a second-half comeback to beat 10-man West Ham 3-2, with two goals from Leighton Baines before Romelu Lukaku headed in the winner.
The meeting of the league's two winless sides ended with West Bromwich Albion beating Sunderland 3-0.
Hull came from behind to win 3-2 at Newcastle, and Aston Villa ended a three-game losing streak by overcoming Norwich 1-0.
At Stamford Bridge, the end of the scoreless first half against Fulham was greeted with boos by some Chelsea fans fearful of a third successive loss.
But Oscar stabbed the ball into the net seven minutes into the second half after goalkeeper David Stockdale parried from Andre Schurrle and spilled Samule Eto'o's shot.
The second goal came from the unlikely source of Mikel, who had never scored in the league since joining Chelsea in 2006.
It came from the corner after goalkeeper David Stockdale pushed striker Fernando Torres' header over, with John Terry nodding the ball onto Mikel to slice the ball into the net.
Liverpool has been the surprise package this season as it searches not only for a first English title since 1990, but just a return to the top four Champions League places.
But against Southampton, Liverpool couldn't find a way past goalkeeper Artur Boruc, who made a key save from captain Steven Gerrard's free kick in the first half and also denied Victor Moses.
Southampton's winner came in the 54th, with Dejan Lovren rising to meet Adam Lallana's corner and head home on the turn.
Liverpool was upstaged by Merseyside neighbour Everton.
West Ham led at halftime through Ravel Morrison and, although Baines equalized in the 62nd, the hosts' lead was restored from the penalty spot by Mark Noble.
The game swung back in Everton's favour after Kevin Nolan was sent off in the 81st for a foul on Ross Barkley just outside the box.
And Baines levelled again for Everton from a free kick two minutes later before substitute Lukaku headed in from Kevin Mirallas' cross.
Lukaku is on loan at Goodison Park after Mourinho found no need for him at Chelsea this season.
Last season he thrived on loan at West Brom.
Without the striker, the Midlands club had taken more than a month into the season to win a league game.
There was a debut goal against Sunderland for Stephane Sessegnon, who was sold by visiting manager Paolo di Canio in an acrimonious deadline day move earlier this month.
After Liam Ridgewell scored the second in the second half, Morgan Amalfitano rounded off the victory in stoppage time.
At Carrow Road, Villa goalkeeper Brad Guzan saved a penalty from Robert Snodgrass in the 6th after Ciaran Clark handled.
It proved costly as Libor Kozak netted for Villa on the half-hour.
The Czech Republic striker, who came on for the injured Christian Benteke, scored with a close-range shot set up by Gabriel Agbonlahor.
At Newcastle, Loic Remy put the hosts in front twice in the first half but the first goal was cancelled out by Robbie Brady and the second by Ahmed Elmohamady.
Sone Aluko volleyed in the winner in the 76th for Hull to pick up its first points on the road since returning to the Premier League.