Champions Chelsea slumped to a shock 2-1 home defeat against Crystal Palace on Saturday with manager Jose Mourinho suffering only his second Premier League defeat at Stamford Bridge.
Meanwhile, Manchester City maintained their perfect start to the season with a 2-0 victory over promoted Watford.
Mourinho was taking charge of his 100th home fixture in England's top-flight, but Palace ensured that milestone won't be remembered with any fondness by the Portuguese coach as they inflicted his first league loss on home soil since Sunderland's victory in April 2014.
It was the latest setback for Mourinho's side, winners only once in their first four matches and already languishing eight points behind leaders Manchester City, who extended the last 100% record in the top tier with a win at Eastlands.
"Palace gave us a very difficult game but the result should be a draw, which would still be bad for us," Mourinho said,
"We created lots of chances but we had a couple of players who were not in the game."
Just seconds after clearing a Gary Cahill header off the line, Mali forward Bakary Sako struck at the other end, sweeping high into the net from Yannick Bolasie's cross to give Palace a 65th minute lead.
Colombia's Radamel Falcao looked to have rescued a point for Chelsea when the on-loan Monaco striker headed in a Pedro Rodriguez cross for his first goal for the Blues in the 79th minute.
However, parity was short-lived as defender Joel Ward headed home from Sako's cross in the 81st minute to give second placed Palace their first win at the Bridge since 1982.
At Eastlands, Raheem Sterling's first goal for City since his 49 million-pound ($75.4 million, 67.4 million euros) pre-season switch from Liverpool broke Watford's stubborn resistance in the 47th minute as the England winger stabbed in Bacary Sagna's cross.
Fernandinho bagged City's second goal in the 56th minute, the Brazilian lashing home after a one-two with David Silva to seal a club record 10th successive league win dating back to last season.
"You can't expect to win the game inside 45 minutes against a team who defends well. We made some changes and we created more chances and won the game," Pellegrini said.
Liverpool were beaten for the first time this season as West Ham's 3-0 victory gave the east Londoners their first league triumph at Anfield since 1963.
In the third minute, Martin Skrtel's poor headed clearance fell at the feet of Aaron Cresswell and his low cross was prodded in by Manuel Lanzini, a 22-year-old Argentine midfielder on loan from Al Jazira.
Slaven Bilic's team doubled their lead in the 29th minute when Mark Noble side-footed home after Reds defender Dejan Lovren carelessly allowed Lanzini to seize possession.
Brazilian playmaker Philippe Coutinho was sent off for his second booking in the 56th minute.
Referee Kevin Friend also sent off Noble in the closing stages, but there was still time for Diafra Sakho to complete Liverpool's embarrassment with a third goal in stoppage-time.
"The first goal is a cheap goal and that gave West Ham motivation and fight. The second was a mistake, which can happen," Reds boss Brendan Rodgers said.
Arsenal clinched their second league win of the season as Fabricio Coloccini's own goal gifted the Gunners a scrappy 1-0 success at 10-man Newcastle.
The turning point came in the 16th minute when referee Andre Marriner sent off Aleksandar Mitrovic after the Newcastle striker needlessly stamped on Francis Coquelin's ankle as he stretched in a failed attempt to reach a loose ball.
Wenger's team eventually took advantage of their numerical superiority in the 52nd minute when Alex Oxlade-Chamberlain's low drive beat goalkeeper Tim Krul via a deflection off Coloccini.
Leicester left it late to rescue a 1-1 draw at Bournemouth thanks to Jamie Vardy's 86th minute penalty after the hosts went in front through striker Callum Wilson's acrobatic overhead kick.
Sunderland remain without a league win despite a battling 2-2 at Aston Villa.
West Bromwich Albion won 1-0 at nine-man Stoke, who had Ibrahim Afellay and Charlie Adam sent off, as Venezuela striker Salomon Rondon scored his first goal since his £12 million club record move from Zenit St Petersburg.
In the late game, Tottenham's 0-0 draw against Everton at White Hart Lane left the hosts still waiting for their first win of the season.
That stalemate also meant there have been only seven wins by home teams in 38 Premier League fixtures this season.
On Sunday Manchester United go to Swansea, while Southampton face Norwich.