Former skipper Tevez, who in midweek was handed his first start since September after ending a dispute with manager Roberto Mancini, produced a scintillating display to fuel his side's title hopes and put pressure on United.
Mancini's men have 77 points with four matches remaining, including a potentially title-deciding home game with United who have 79 from one game fewer.
Thanks to their commanding win at Norwich, Manchester City's goal difference is now eight better than United's, the team they walloped 6-1 at Old Trafford during a barnstorming start to the season.
United, whose shock midweek defeat at Wigan Athletic combined with Manchester City's home win against West Bromwich Albion rekindled the title race, host Aston Villa on Sunday.
Impressive as Manchester City were, a rather subdued Mancini refused to talk up his club's hopes of winning their first top-flight title since 1968.
"It was a fantastic performance," the Italian told Sky Sports. "We need to finish well in the championship and we are starting to play very well.
"But when you fight against a fantastic team like United it's impossible. (The title race) is finished.
"I think it is too late. I am happy we are playing like we did in the first 28 games."
At the other end of the table Blackburn Rovers's hopes of avoiding relegation suffered a big blow after a 3-0 defeat at Swansea City.
Bottom club Wolverhampton Wanderers look almost certain to slip out of the top flight after a 0-0 draw at Sunderland left them eight points from safety.
Queens Park Rangers, two points above the drop zone, went down 1-0 at West Bromwich.
Blackburn manager Steve Kean said all was not lost, despite his second from bottom team falling three points behind QPR and Wigan who meet third-placed Arsenal on Monday.
"It is obvious what we have to do. We have to win our next game. We'll have to win two games and then get a draw from somewhere. It's as simple as that," Kean told the BBC.
Tevez, who has missed most of the season after refusing to warm up in a Champions League match at Bayern Munich and then going back to Argentina without the club's consent, could yet propel Manchester City to the title.
After a few substitute appearances and an hour from the start against West Bromwich in midweek, he looked back to his best against a Norwich side who played their part in a lively match.
The mercurial striker opened the scoring after 18 minutes with a wickedly dipping shot that beat flapping keeper John Ruddy and his clever back heel nine minutes later allowed Aguero to smash home Manchester City's second.
Norwich threatened a revival when Andrew Surman shot through a crowd of players after 51 minutes but the visitors weathered a few anxious moments and then romped to victory.
Tevez nodded the third after 73 minutes when Yaya Toure's shot bounced back off Ruddy, Aguero curled in his second two minutes later and Tevez completed his hat-trick by almost walking the ball into the net after a defensive mistake.
Aguero almost matched his strike partner's goal tally late on when he thumped a shot against the woodwork before Adam Johnson completed the rout in stoppage time.
"Enough has been said, all we had to do was go out and play," Manchester City skipper Vincent Kompany told Sky Sports.
"We realised that for a couple of months we did not perform at our best and today with Carlos and Sergio up front it was not just about the goals, we were fired by their performance and work rate."
