Chelsea's problems worsened as Southampton battled back to beat the struggling Premier League champions 3-1 at Stamford Bridge.
Willian bent a free kick into the top corner from wide on the left after 10 minutes -- but that was as good as things got for Jose Mourinho's struggling side.
Southampton equalised shortly before the break when Graziano Pelle chested the ball into the path of Steven Davis, who smashed a low drive past Asmir Begovic from the edge of the box.
Begovic made a fine save from Sadio Mane early in the second half, but the Southampton attacker got the better of him on the hour when Gary Cahill gave possession away, and then John Terry failed to cut out Pelle's pass.
And Pelle wrapped up an outstanding victory for the Saints when, after Chelsea had lost possession again, he fired home from the right-hand side of the area.
There were boos from the home fans when substitute Nemanja Matic was substituted and further discontentment at the full-time whistle as Chelsea's early-season worries deepened further.
Sergio Aguero scored five goals as Manchester City came from behind to thrash Newcastle 6-1 at the Etihad Stadium.
City end the day back on top of the Premier League after a resounding victory in which Aguero scored five in 20 minutes during a remarkable spell that also saw Kevin De Bruyne net for the hosts.
Newcastle, who drew 2-2 with Chelsea last weekend after having had a two-goal lead, went ahead in the 18th minute when Aleksandar Mitrovic stooped to head home a Georginio Wijnaldum cross for his first goal since his summer move from Anderlecht.
He saw a second goal ruled out for offside on 25 minutes, but City were level just before the break when Aguero beat Tim Krul with a diving header from Fernandinho's headed assist.
That sparked a goal glut that saw Aguero complete his hat trick by the 50th minute with De Bruyne producing an acrobatic finish for his side's fourth.
Aguero added two more goals in the 60th and 62nd minutes to match the Premier League record for goals in a single match before being substituted by manager Manuel Pellegrini four minutes later.
Sunderland threw away a two-goal lead as they were held to a 2-2 draw by West Ham United at the Stadium of Light.
Sunderland, who had taken just two points from their opening seven games this season, went into the game on the back of strong speculation that coach Dick Advocaat had decided to walk away regardless of the result.
However, Advocaat was given cause to reconsider as Steven Fletcher fired home following a Yann M'Vila corner on 10 minutes before Jeremain Lens chipped home a second on 22 minutes.
Fabio Borini missed a glorious chance to make it 3-0 on 38 minutes, and his failure to hit the target was punished just before the break when Carl Jenkinson pulled one back for the Hammers.
Sunderland then saw Lens sent off when he picked up a second booking on 57 minutes, and West Ham were level on the hour as Dimitri Payet fired home.
Leicester City continued their impressive start to the season with a 2-1 victory at Norwich City.
England international Jamie Vardy gave the Foxes the lead when he won and converted a penalty for his sixth goal in eight Premier League appearances.
Leicester doubled their advantage shortly after the restart when Jeffrey Schlupp scored on the counter-attack, but Dieumerci Mbokani gave the hosts hope when he flicked home a Jonny Howson cross on 68 minutes.
However, Leicester held on for a victory that leaves them fourth in the table.
Aston Villa's poor start to the season continued as they were beaten 1-0 at home by Stoke City.
Stoke scored the only goal of the game on 55 minutes when Glen Johnson picked out Marko Arnautovic, who fired a low shot past Brad Guzan.
Bournemouth were denied victory by a late penalty save as they drew 1-1 at home to Watford.
Glenn Murray gave the Cherries the lead on his full debut for the club, heading past Heurelho Gomes on 28 minutes, but Watford were gifted an equaliser just before half-time when keeper Artur Boruc gave the ball away to Odion Ighalo for a simple finish.
Murray had the chance to win it from the penalty spot late on, but Gomes made the save to give the Hornets a share of the spoils.
Crystal Palace continued their impressive start to the season with a 2-0 home win over West Bromwich Albion in Saturday's early game.
Yannick Bolasie made the breakthrough when he headed home Yohan Cabaye's superb cross from the left after 67 minutes, and Palace doubled their advantage when Wilfried Zaha won a late penalty, which Cabaye converted.