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 npower Championship, while United remain just outside the relegation zone.
Source: PA
Source: PA