The Canaries were second-best for much of the first half at Bramall Lane against a struggling United side still searching for their first win under Micky Adams.
The Blades once again lacked a cutting edge in attack, however, and were made to pay for their wastefulness in front of goal after the break as Crofts fired home either side of Ched Evans’ 69th-minute strike.
Crofts opened the scoring in the 61st minute after Steve Simonsen had flapped at a David Fox corner, although the hosts felt that Kyle Bartley had been fouled by Simeon Jackson in the attack that led to the set-piece.
United quickly got back on level terms as Evans turned his marker and squeezed a right-footed shot into the bottom-left corner.
But Crofts struck again with 10 minutes remaining to seal victory for Norwich, the midfielder again finding the net from inside the area after the home side had failed to clear their lines.
The win moves Paul Lambert’s side up to third in the Championship, while United remain just outside the relegation zone.
Marcus Bent made his second debut for United after rejoining the club on loan from Birmingham this week. Richard Cresswell was also handed a starting berth, with Johnny Ertl and Lee Williamson dropping to the bench.
There was one change to the Norwich side that drew 1-1 with Cardiff last weekend, Jackson coming in for Henri Lansbury.
Canaries goalkeeper John Ruddy was forced into action in the sixth minute when Jamie Ward flicked Joe Mattock’s left-wing cross towards the bottom-right corner.
Grant Holt then saw a header sail over for the visitors, while at the other end Evans blasted a 25-yard half-volley narrowly wide after strike partner Bent had nodded a long ball into his path.
Evans had a better opportunity to open the scoring in the 17th minute as he cut into the penalty area from the right, but his left-footed shot lacked power and was comfortably saved.
Ruddy denied the striker once more five minutes later, sticking out his left leg to divert the ball over after Evans had chested down a cross from the left and shot powerfully at goal.
Chances were few and far between for the remainder of the half, although Evans did go close again in the 34th minute as he curled a free-kick narrowly wide from 25 yards.
Holt and Stephen Quinn were both lucky to escape bookings after they were involved in an unsavoury tussle towards the end of the first half.
Norwich had offered next to nothing as an attacking force, but came closest to breaking the deadlock on the stroke of half-time when Chris Martin’s free-kick struck the crossbar.
Richard Cresswell fired over for United and Holt volleyed into the arms of Simonsen in a lively start to the second period.
Shane Lowry produced a fine block to deny Jackson on the hour after the striker had got the better of Bartley, but Norwich took the lead from the resulting corner as Crofts capitalised on Simonsen’s error.
Evans finally found a way past Ruddy eight minutes later following a neat turn inside the area.
But it was to be an improving Norwich who finished the stronger, with Martin forcing a sharp stop from Simonsen and Holt heading wide prior to Crofts’ late winner.