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.
City, who suffered their first league defeat of the season last week at Chelsea, now have 41 points from 16 matches, two ahead of United.
Tottenham Hotspur kept up the pressure, beating Sunderland 1-0 at White Hart Lane to take third place on 34 points with a game in hand on the leaders. Roman Pavlyuchenko grabbed the winner in the 61st minute.
Chelsea, who drew 1-1 at relegation-threatened Wigan Athletic on Saturday, fell to fourth on 32 points with Arsenal fifth on 29 and ahead of Liverpool, 2-0 winners at Aston Villa, on goal difference.
City, with Samir Nasri and Kolo Toure facing their former team-mates in an intense and compelling clash between two great passing teams, knew what they had to do.
"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.
At the Arsenal end, Aguero missed an early chance to score while Argentine defender Pablo Zabaleta hit the post with a fierce drive.
"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."
Mancini said he had congratulated his players in the dressing room but had also criticised them for failing to control the game and leaving themselves open to counter-attacks.
Arsenal manager Arsene Wenger, who had to reshuffle his defence after the break when Johan Djourou was injured, said it was always difficult to take defeat after such a game.
"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
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