After failing to score during its first-half supremacy, Arsenal capitulated with defender Kieran Gibbs scoring an own goal and Wayne Rooney netting for United. Olivier Giroud's goal was too late to prevent United winning away in the league for the first time this season, sending Arsenal tumbling to eighth in its worst start for 32 years.
"Defensively, we were a bit naive," Arsenal manager Arsene Wenger said.
Chelsea, though, maintained its firm grip on top spot with a serene 2-0 victory over West Bromwich Albion secured by Diego Costa and Eden Hazard.
City remains eight points back in third place — five ahead of United — after being forced to come from behind against Swansea. Stevan Jovetic cancelled out Wilfried Bony's opener for Swansea before Yaya Toure clinched the champions' 2-1 win in the second half.
Newcastle's resurgence after a woeful start to the season continued with a fifth straight league win. Moussa Sissoko scored in the 1-0 victory over Queens Park Rangers which has left Newcastle fifth while the London club went bottom.
Burnley climbed off the foot of the standings by recording its first away triumph, with Danny Ings' double claiming a 2-1 victory at Stoke.
West Ham's five-match unbeaten run ended with a 2-1 loss to Everton, while Leicester's goalless run went beyond 500 minutes after drawing 0-0 with Sunderland.
Why Arsenal didn't score in the first half at the Emirates Stadium perplexed Wenger. The hosts were in control, peppering United goalkeeper David de Gea with shots before imploding in the 56th minute.
When Ashley Young whipped in a cross, Gibbs clattered into his own goalkeeper, Wojciech Szczesny. United winger Antonio Valencia crossed the loose ball back into the penalty area, and Gibbs stuck out a leg and diverted the ball into his own goal.
Szczesny was forced off with an apparent rib injury and was replaced by Emiliano Martinez. The third-choice goalkeeper was beaten in the 85th after Rooney raced onto Angel di Maria's pass.
"We weathered the storm at times, especially first half," Rooney said. "Then we knew we could hit them on the counterattack, we knew they were vulnerable on the break."
Chelsea's victory was never in doubt. Costa volleyed in the opener in the 11th minute after meeting Oscar's cross, and Hazard met Cesc Fabregas' corner in the 25th.
The only surprise for Chelsea was the failure to record a more impressive victory margin after Claudio Yacob saw red for a challenge on Costa in the 29th.
"They knew that the points were not at risk," Mourinho said, adding: "Many, many times I am upset, but I think not today."
Neither was Manchester City manager Manuel Pellegrini thanks to the champions' comeback, having raised the tempo after falling behind.
Bony skillfully put Swansea in front, using his chest to control Nathan Dyer's ball before sending the shot past goalkeeper Joe Hart in the 9th. But Jovetic levelled 10 minutes later, diverting Jesus Navas' cross past goalkeeper Lukasz Fabianski.
City equalized when Toure pushed forward and fed Jesus Navas out wide. The Spaniard curled an inviting low ball into the area and Jovetic stuck out a leg to divert past Lukasz Fabianski.
And Toure scored just his second league goal of the season in the 62nd after combining with Fernandinho.
Second-placed Southampton is at Aston Villa on Monday.