United, who suffered their third Premier League defeat of the season in the late kick-off at Carrow Road, succumbed to a 60th-minute header from their former youth team player Anthony Pilkington.
Chelsea lost 2-1 at West Brom, with Peter Odemwingie's second-half winner sending the home side into fourth place and ruining the return of former Baggies manager Roberto Di Matteo.
City's victory at the Etihad Stadium, engineered with two goals apiece from Argentine duo Sergio Aguero and Carlos Tevez, took them top of the league for the first time this season.
Roberto Mancini's side have 28 points from 12 matches, with United second on 27 points. Chelsea have 24 points from 12 games with Albion a point further back.
Arsenal had kickstarted their season in the early Saturday kick-off by beating North London rivals Tottenham Hotspur 5-2 after Spurs striker Emmanuel Adebayor was sent off for a reckless first-half challenge on Santi Cazorla.
In-form Luis Suarez scored twice to help Liverpool to a 3-0 win at home to Wigan Athletic, who lost midfielder Ben Watson to a broken leg, while Swansea City enjoyed a 2-1 victory at a struggling Newcastle United.
Southampton increased the pressure on Queens Park Rangers manager Mark Hughes by winning their basement battle 3-1 away and Reading - who also started the day in the bottom three - surprised Everton with a 2-1 home win, Adam Le Fondre's double leading them to their first league win this season.
MISSED ROONEY
United missed Wayne Rooney, who was sidelined with tonsillitis, though Sir Alex Ferguson was still able to pair Robin van Persie with Javier Hernandez, who had claimed a hat-trick seven days ago as United came back from 2-0 down to beat Aston Villa 3-2.
Ferguson's men were suddenly facing another deficit when Pilkington got in front of defender Chris Smalling to glance a deft header from Javier Garrido's cross past keeper Anders Lindegaard.
Ferguson had said this week he could not remember Pilkington from his time at Old Trafford, though the midfielder joked after the game: "I was only about five, so that's probably why."
United were so poor they only put John Ruddy under serious pressure in the dying moments, the Norwich goalkeeper doing well to turn away a misplaced header from defender Sebastien Bassong and then saving bravely at the feet of Paul Scholes after he had blocked Anderson's 20-metre drive.
"We're disappointed, but they deserved it," Ferguson told ESPN. "We had one or two half-chances, and the players are used to making comebacks, but Norwich defended really well."
West Brom fans gave Chelsea manager Di Matteo a warm welcome on the Italian's return to the Hawthorns and then their team showed him how far they had come since he was sacked in 2011.
Shane Long headed the home side ahead in the 10th minute and after Eden Hazard equalised with a header six minutes before the break, it was another header that gave West Brom the points with Long this tim
