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United maintained their seven-point advantage over City at the top of the Premier League after Gareth McAuley turned the ball into his own net and substitute Robin van Persie netted in a 2-0 win over West Bromwich Albion at a sodden Old Trafford.
City striker Dzeko scored twice in the first four minutes and another of his strikes was credited as an own goal by goalkeeper Mark Bunn at Norwich where Samir Nasri's 44th-minute red card had threatened the champions' dominance.
"We played fantastic for 25 minutes and should have scored three or four. I don't know why (Nasri was sent-off) as the linesman was there and I don't know what he saw. We saw the video, both players touch heads," City boss Roberto Mancini told the BBC.
"Today we showed we are there for the title."
In-form Spurs moved third following a 2-1 comeback win at Sunderland where an apoplectic Gareth Bale was again booked for diving while fourth-bottom Aston Villa lost their third on the bounce in a 3-0 home defeat by Wigan Athletic.
Paul Lambert's Villa have conceded 15 in the three games.
Ten-man Stoke City were facing a first home loss since February but a superb late Cameron Jerome strike secured a 3-3 draw with struggling Southampton and Reading pulled themselves away from last spot with a 1-0 home win over West Ham United.
Swansea City shrugged off the injury absence of top scorer Michu to triumph 2-1 at Fulham where stand-in striker Danny Graham grabbed the opener.
Leaders United started with Van Persie on the bench and had Wayne Rooney out injured for a game that was nearly called off before kickoff because of persistent rain.
Japan midfielder Shinji Kagawa made his comeback from injury but it was another returnee, winger Ashley Young, who provided the cross for McAuley to inadvertently score on nine minutes.
Alex Ferguson's side have been consistently leaking goals, only prevailing 4-3 over Newcastle United on Wednesday thanks to a last-minute goal, but they just about held on to a clean sheet before Van Persie eased fan nerves in the last minute.
After 20 of 38 games, United have 49 points to City's 42 but Mancini will be glad his champions did not slip further behind after playing with 10 men for over a half at Carrow Road.
Bosnian forward Dzeko made a stunning start for City, whose forwards had previously scored just three goals in eight games, but Anthony Pilkington's 15th-minute goal and Nasri's debatable sending off left the game in the balance.
France midfielder Nasri saw red for the first time in England for going head-to-head with Sebastien Bassong but the dismissal seemed harsh.
Sergio Aguero then gave City breathing space on 50 minutes before a Dzeko strike came back off the post and hit Bunn on its way into the net, in between Russell Martin's double for spirited midtable side Norwich.
Two goals in three second-half minutes helped Tottenham fight back for a deserved win at Sunderland which sent the north Londoners up to third.
"I think we started well, we went one goal down but were on top of the game really. We showed our character," match-winner Aaron Lennon said.
The hosts, who beat Manchester City 1-0 at home on Wednesday, took the lead against the run of play in the 40th minute when captain John O'Shea coolly tapped in after Spurs had failed to deal with a free-kick into the box.
It was the Irishman's first goal for mid-table Sunderland and his first league goal since netting for Manchester United against Stoke City in September 2009 although his game ended early with a hamstring strain.
Spurs, who thumped Villa 4-0 away on Wednesday, had dominated the first half with Emmanuel Adebayor hitting the bar but they were level three minutes after the break.
Carlos Cuellar headed a corner into his own net and another unlucky deflection on 51 minutes presented Lennon with the chance to make in 2-1 before Bale's third booking for diving this term when he felt he had been fouled by Craig Gardner.
Chelsea can reclaim third spot at Everton on Sunday when bottom side Queens Park Rangers host Liverpool.