Silva's 53rd-minute tap-in at the far post from close range, after Wojciech Szczesny parried a fierce shot by Mario Balotelli and Sergio Aguero nodded on the rebound, settled an end-to-end thriller at the Etihad Stadium.
While Arsenal proved tough nuts to crack in freezing Manchester, where the Londoners posed a constant threat right up to the final whistle, United were 1-0 up after just 52 seconds at promoted Queens Park Rangers thanks to Rooney's diving effort.
Michael Carrick followed up his England team mate's 12th league goal of the season with his first in nearly two years for United to seal a comfortable 2-0 win and knock City off the summit for a few hours.
"We put pressure on ourselves, we don't need anyone else. We watched the game where United did brilliant but we knew we had a job this afternoon and if we won it we were back where we want to be," said City goalkeeper Joe Hart, who made a string of saves including a late tip over the bar from Thomas Vermaelen.
Three minutes after Silva's goal, league top scorer Robin van Persie had the ball in the net but was ruled offside.
"It [the win] was very important after Chelsea," said City manager Roberto Mancini, whose side lost 2-1 at Chelsea last Monday.
"Against Chelsea we didn't deserve to lose the game. Today was a fantastic game... for the supporters because it was open until the end."
"Both teams played in a positive spirit and went for it," the Frenchman said. "It was I felt very intense physically."
Rooney's goal was the third fastest of the season and United could have made it 2-0 after three minutes against opponents
