A dramatic grandstand finish, orchestrated by second half substitutes Wes Hoolahan and Anthony McNamee, ensured the Canaries came from behind twice to end 2010 on a high.
Despite controlling the opening sequences, City fell behind to a 17th minute Andrew Reid free-kick. But they hit back quickly when just three minutes later Michael Nelson headed the equaliser, only for Richard Cresswell to restore the Blades advantage.
The turning point in the game came when Hoolahan converted a 64th minute penalty to draw his side level again. The mercurial Irishman converted another penalty and scored in the 7th minute of extra time to seal an amazing 4-2 victory.
Paul Lambert had had to change his personnel again, with both Elliott Ward (calf) and Henri Lansbury (hamstring) surprise absentees from the side that won at Coventry. He was also missing Leon Barnett, serving the first of a two-match suspension handed out to the central defender by the FA for throwing the ball at the referee in the home defeat to Portsmouth.
City kicked off towards the River End in an unsual 3-4-3 formation, with Michael Nelson and Jens Berthel-Askou either side of Russell Martin in defence.
However, after a third of the first half Lambert acknowledged that his starting line-up wasn't working and reverted to a more conventional 4-3-3 side, with Simon Lappin dropping back from left midfield and Martin moving out to his preferred right-back.
Ched Evans, who whilst on loan at Carrow Road from Manchester City scored 10 goals in 28 games, was fouled just outside the area by Berthel-Askou in the 16th minute.
Reid curled the resulting freekick over the City wall into the far top corner to give John Ruddy no chance. The Blades lead was short-lived as they then conceded from a set-piece. David Fox's freekick from wide on the right went to the far post, where the unmarked Nelson ghosted in to head in the equaliser.
However, for the second time and despite having the lion's share of the possession, City went behind in the 29th minute. Evans was again involved in the build-up that ended with Mark Yeates teeing up Richard Cresswell, who put his low shot to the left of the diving Ruddy.
HALF-TIME: City 1 United 2
The City boss wasted no time in trying to influence the game, starting the second half with Hoolahan replacing the ineffective Simeon Jackson.
Within just two minutes of the re-start, Lappin was booked for a foul on Evans in front of the City stand.
Clear cut chances were still at a premium and in the 61st minute Lambert threw his second roll of the dice, swapping Fox for McNamee.
The referee handed City a lifeline in the 64th minute when he awarded a penalty for a dubious challenge on Holt by Nick Montgomery. Sub Hoolahan didn't make any mistake from the spot to bring the Canaries level and put them in the driving seat.
They created several chances but found the Blades keeper Steve Simonsen equal to anything that Chris Martin, Holt and McNamee fired his way.
Hoolahan put the home side ahead for the first time in the game when he conveted from the penalty spot, after a more clear cut decision when a through ball into the area, was blocked by Kyle Bartley. The Irishman put the 85th minute spot-kick the other way to beat Simonsen for a second time.
The Blades caretaker manager then made his first change, bringing on striker Daniel Bogdanovic for Jean Calve in an attempt to try and salvage something from a game that for over half of it, looked like they were going to win.
With just one minute of normal time remaining, the attacking McNamee was sacrificed for the defensive Zak Whitbread, who made his first City appearance since a calf injury in the home game against Stockport has kept him out since April this year.
Despite the Assistant Referee flagging for a foul on Lappin, the referee was unaware and had it not been for Ruddy's match-winning save from Evans, the end to the game could have been so different!
In the fifth minute of extra time Quinn was shown a straight red card, after a melee of players came together following his challenge left Nelson on the ground!
Then Grant Holt created the icing on top of the cake, when in the dying stages of the game he raced down the right wing and rather than head for the corner flag, he unselfishly crossed to the unmarked Hoolahan to score City's fourth and the Irishman's first Canary hat-trick.
FULL-TIME: City 4 United 2