Champions Chelsea lost 3-1 at Everton before a stoppage-time goal for leaders Manchester City sealed a 1-0 win over Crystal Palace and Manchester United beat Liverpool 3-1 in football's English Premier League.
Manchester United's Anthony Martial celebrates after scoring their 3rd goal. Photo: PHOTOSPORT
Steven Naismith's hat-trick for Everton condemned Chelsea to their third defeat in five games while City avoided dropping their first points of the season when teenage substitute Kelechi Iheanacho struck the winner at Selhurst Park.
Daley Blind and Ander Herrera put United 2-0 up against arch-rivals Liverpool.
Christian Benteke pulled one back with a spectacular overhead kick before recent big money signing Anthony Martial scored a classy goal on his debut to seal victory for the home team.
Manchester City lead the way with 15 points, five clear of second-placed United and Arsenal, who are third after beating Stoke City 2-0, and 11 ahead of struggling Chelsea.
Manchester City were pushed hard by Palace, who started the day in second spot, and looked set to be denied their fifth successive win until Iheanacho came off the bench to grab the only goal.
"We knew it was going to be a tough game with the first team in the table playing the second," City manager Manuel Pellegrini said.
Chelsea boss Jose Mourinho remained in defiant mood after his team's third defeat of the season, the same number of games they lost in the whole of last term.
"They are a very good team, in a good moment, so it was a very important three points," said Mourinho.
"But we dominated the second half and had the better chances. The results are too wrong for our quality and our status and the players deserve much more than they are getting."
Mourinho said he was the best man to turn around Chelsea's wretched start.
"I don't blame the players and I don't blame myself. They are the worst results ever in my career but I am comfortable with the situation," he added.
"I do not think there is a better manager than me to come to Chelsea and do my job.
"I am the man for the job. I have done so many good things for Chelsea. Now I am the man with the bad results but give me the responsibility and I will sort this out."
Liverpool manager Brendan Rodgers described the game against United as the biggest derby in English football but the match only came to life in the second half after a dire opening 45 minutes.
The catalyst was a well-worked 49th-minute free kick that Blind curled home before Herrera hammered in a 70th-minute penalty to put United 2-0 ahead.
Benteke gave Liverpool late hope but his effort was over-shadowed by Martial's stunning run and shot which bamboozled three defenders and keeper Simon Mignolet, showing why United spent a world record $88 million for the teenager earlier this month.
"The goal from Anthony, not a bad goal," United manager Louis van Gaal said.
"He did alright, he had some space because Liverpool had to come out. He scored a marvellous goal. I've seen that physically he can play in the Premier League."
Arsenal grabbed their first home win of the season against Stoke thanks to strikes from Theo Walcott and Olivier Giroud.
Norwich City beat Bournemouth 3-1 and promoted Watford sank Swansea City 1-0, Odion Ighalo's second-half strike securing their first league win of the season.
The victory was tarnished when Valon Behrami was sent off after appearing to stamp on Andre Ayew in the 64th minute.
West Bromwich Albion and Southampton drew 0-0.