Paul Scholes had given the visitors an early lead in Sunday's Premier League clash with a close-range header.
The Canaries rallied and forced David de Gea into a string of fine saves before the United number one was finally beaten with seven minutes left when Grant Holt fired home from 12 yards.
United, though, were not to be denied as they poured forwards in the closing stages and were eventually rewarded when Giggs stabbed home from close range before celebrating wildly in front of the travelling fans.
Sir Alex Ferguson had recalled his regulars following their midweek defeat by Ajax in the Europa League, looking to close back up on leaders Manchester City, although England striker Wayne Rooney missed out because of a virus.
The Canaries have consolidated themselves well on a first return to the top flight in the six seasons since last facing the Red Devils here, securing back-to-back promotions up from npower League One under Paul Lambert.
Only three teams have won at Carrow Road in the Premier League this season, and it was another positive start by the hosts as Simeon Jackson looked to get in between centre-backs Rio Ferdinand and Jonny Evans.
United, though, soon settled and retained possession well, and were in front on seven minutes.
Nani swung the ball over from the right, and it bounced through the six-yard box to the far post, where Scholes nodded it past John Ruddy.
Bradley Johnson was shown a yellow card by referee Andre Marriner for a crunching tackle on the Portuguese winger, which on another day could well have been red.
On 15 minutes, it was almost 2-0 when another flowing United counter opened up the Norwich defence before Danny Welbeck's scuffed shot was touched wide at full stretch by Ruddy.
Norwich finally created a meaningful chance on 28 minutes, Kyle Naughton chipped a throughball into the right side of the United penalty box. Anthony Pilkington took it in his stride with a sublime first touch before knocking his shot goalwards, with De Gea making a fine reaction save at the near post.
Javier Hernandez should have done better when played clear into the Norwich penalty area, but failed to trouble Ruddy with a tame effort straight at the keeper.
At the other end, De Gea was producing a top-drawer display as he plucked a glancing header from Holt out of the air after the Norwich captain turned Pilkington's right-wing cross goalwards.
At the start of the second half, Giggs weaved his way into the Norwich box and tumbled under a challenge from Zak Whitbread, but play was waved on.
On 57 minutes, Johnson's low, 25-yard effort floated just wide.
Pilkington's looping cross fell to Andrew Surman at the far post, but his well-executed volley was blocked.
Welbeck and Giggs exchanged passes to send the striker clear, and his chip floated past Ruddy, but Whitbread was alert to recover and hack clear even though the ball may have been going wide.
Norwich maintained their pressure, with Bennett's cross from the right pluck out of the air by De Gea as Holt lurked in the six-yard box.
Giggs was denied a goal to cap his milestone when his close-range volley cannoned off the top of the bar from Phil Jones' cross on 72 minutes.
Wes Hoolahan's cross then hit Evra on the elbow as Norwich screamed for a penalty, but referee Marriner thought otherwise.
Scholes gave the ball away to substitute Aaron Wilbraham 25 yards out, and the big striker strode forwards before unleashing a shot towards the top corner, which De Gea turned over at full stretch.
Norwich, though, were not to be denied as the resulting corner was cleared back to Adam Drury, who launched it into the area again. Whitbread knocked the ball down to Holt, who squirmed between two markers before firing in his 10th league goal of the season on 83 minutes.
Young's goalbound effort was blocked and Welbeck somehow missed a free-header from two yards.
However, Giggs ensured a fairytale finish when he stabbed home in stoppage time to keep United on the tails of leaders City.