McNulty sent the Blades through after Alex Russell levelled for Mansfield just four minutes after Paul Butler's opener.
United's Jose Baxter had an early shot beaten away by goalkeeper Sascha Studler while Neill Collins and Butler both headed off-target.
Mansfield threatened when Simon Heslop made a good run at the defence and laid the ball off to Sam Clucas whose low shot was saved by Mark Howard.
Heslop saw an effort deflected wide soon after the re-start and a header from Clucas flicked the top of the bar before Butler headed home at the far post in the 53rd minute after Davies' corner caused problems for Studler.
But Fisher drew the League Two side level with a shot from inside the area after Lee Beevers' effort had struck a defender.
The game looked to be heading for extra time until McNulty grabbed his first Blades goal in the 86th minute to give the home side victory. The forward held off Ritchie Sutton's challenge inside the area before firing hard and low past Studler.
The result means United will play West Ham in the second round - the first time since the two sides have met since the controversy over Carlos Tevez when his goal kept the London side in the Premier League at the Blades' expense in 2007 - Tevez was subsequently revealed to have been under third-party ownership.