On a high-scoring final Saturday of 2012, Gareth McAuley's own goal and a league-high 14th strike of the campaign by substitute Robin van Persie earned United a seventh straight home win to leave City with a mountain to climb to retain its title.
"We're in a good position. We're halfway there,'' United manager Alex Ferguson said.
The reigning champions withstood the first-half dismissal of playmaker Samir Nasri for an apparent headbutt to hold off Norwich, with Edin Dzeko (2) and Sergio Aguero answering manager Roberto Mancini's call for more goals from his strikers in a thrilling game at Carrow Road.
"Today we showed we are there for the title,'' Mancini said.
Tottenham came from behind to beat Sunderland 2-1 and move provisionally third _ a point ahead of Chelsea, which visits Everton on Sunday.
The win wasn't without controversy, though, with winger Gareth Bale booked for diving for the third time this season _ and fifth time since the start of last season.
"If people kick me, I'm going to go down,'' Bale said in protest, despite going down easily this time when Sunderland defender Craig Gardner put his arm across Bale's chest. ``People keep saying I'm diving but it doesn't bother me whatsoever.''
At the other end of the standings, Reading climbed three points clear of last-place Queens Park Rangers by beating West Ham 1-0 thanks to Pavel Pogrebnyak's fifth-minute goal. It was only Reading's second win of the season.
Southampton slipped into relegation zone despite drawing 3-3 at Stoke, which equalized in the last minute through a wondrous goal by Cameron Jerome, because Wigan piled on the misery for Aston Villa by winning 3-0 at Villa Park.
Villa has now conceded 15 goals _ and failed to score _ in three matches over the festive period to lie just a point and a place above the bottom three.
Fulham's poor form continued with a 2-1 home loss to Swansea, leaving the hosts with just one win in 12 league games.
Arsenal hosted Newcastle in the other game later Saturday, while QPR is at home to Liverpool on Sunday.
United's latest win at Old Trafford _ coming two days before Ferguson's 71st birthday _ lacked the drama and controversy of the last-gasp 4-3 victory over Newcastle on Wednesday, with driving rain and a boggy pitch making for tough conditions. Indeed, a pitch inspection was needed before the match was given the go-ahead less than two hours before kickoff.
The goals came at either end of the match, with McAuley deflecting in Ashley Young's low cross in the ninth minute and Van Persie coming off the bench to curl home the clinching goal in the 90th minute. That took the Dutchman one goal ahead of Swansea forward Michu in the scoring charts.
"He (Van Persie) changed the game for us,'' Ferguson said. ``When I brought him and Paul Scholes, on they settled the game down for us.''
City's win was more nerve-racking, with Nasri's straight red card in the 44th minute with the score at 2-1 to the champions giving Norwich hope.
The France playmaker motioned a headbutt to Sebastian Bassong as the two squared up to each other and clashed foreheads. While Bassong was given a yellow card, Nasri was dismissed.
"Samir said he touched his head, OK, but the other player has done the same,'' said Mancini, who added that he would fine Nasri but also appeal against the decision.
Dzeko's brace in the opening four minutes put City 2-0 ahead but Anthony Pilkington's free kick halved the deficit in the 15th.
Down to 10 men, City's two-goal lead was restored five minutes after halftime by Aguero. Russell Martin to make it 3-2 in the 63rd and then 4-3 in the 75th after Norwich goalkeeper Mark Bunn had deflected in Dzeko's low drive for an own goal _ the fourth of the day in the Premier League _ but City held on.
Spurs are City's closest challengers _ at least until Sunday _ as an own goal by Carlos Cuellar and Aaron Lennon's fine individual strike in a five-minute span at the start of the second half gave them a come-from behind win at the Stadium of Light.