The Blades, with just one win in their last eight Coca-Cola Championship matches, appeared to have got rid of their jitters when Chris Armstrong fired them into a 17th-minute lead.
But Leon McKenzie stole in to snatch an equaliser following a set-piece and just seconds after visiting striker Neil Shipperley had seen his 25-yard shot smash back off the post, Robert Earnshaw blasted home from close range to put Norwich in front before the break.
United dominated after the restart but found England World Cup hopeful Robert Green in inspired form, with second-half saves from Michael Tonge and Danny Webber in particular catching the eye.
Norwich manager Nigel Worthington recalled Earnshaw to his starting line-up in place of Simon Charlton, while opposite number Neil Warnock made three changes and reverted to a 4-4-2 formation.
Defenders Rob Kozluk and David Unsworth returned to the back four following last week's Coca-Cola Championship defeat at Coventry and Phil Jagielka was pushed up into midfield alongside Tonge, who returned following suspension.
Earnshaw almost made an instant impact, flashing a 25-yard volley wide after just seven seconds, but neither side created another chance until the visitors opened the scoring.
Tonge fed Webber on the edge of the penalty area and when the ball fell for Armstrong following Jason Shackell's tackle, the Blades defender jinked inside and drilled a left-footed shot into the bottom corner.
United threatened again soon after and City were grateful for Shackell's intervention when he blocked Jagielka's goalbound volley.
Norwich had not carved out one worthwhile effort on goal until they equalised out of the blue in the 33rd minute.
Adam Drury swung over a free-kick which Shackell headed on into the six-yard area and nobody picked up the diminutive McKenzie, who pounced to flick the ball beyond Blades goalkeeper Paddy Kenny.
Nick Montgomery and Webber pulled shots wide and Earnshaw's lob bounced the wrong side of goal before Shipperley's thunderbolt hit the post and Earnshaw grabbed Norwich's second following Craig Fleming's cross.
Armstrong volleyed wide from Tonge's cross soon after the restart, with the Blades looking full of purpose in pursuit of an early equaliser.
Craig Short stabbed the ball wide following Unsworth's corner and Webber's strike was blocked by Shackell, but after soaking up early second-half pressure Norwich almost extended their lead.
Darren Huckerby stole in on the left and unleashed a right-footed shot that Kenny did well to parry for a corner.
Home goalkeeper Robert Green then twice denied the visitors in fine style, first holding onto Tonge's stinging 20-yard shot and then acrobatically catching Webber's similar effort to keep the Blades at bay.
Jonatan Johansson replaced McKenzie in the 69th minute with fellow substitute Youssef Safri's shot blocked by Montgomery before Shackell's header from Huckerby's free-kick hit the crossbar.
Warnock's last throw of the dice saw Ade Akinbiyi and Keith Gillespie replace Shipperley and Montgomery with 10 minutes remaining, but although the crosses rained in on Norwich's penalty area, the Blades could not find that elusive equaliser.