United cracked in three goals in an 18-minute spell after lowly Stockport County had taken Neil Warnock's side into extra-time.
Alan Quinn struck first in the 108th minute as he cracked in a shot from 25 yards, the ball cannoned off the post, hit keeper Neil Cutler and rolled over the line.
Substitute Michael Tonge added United's second four minutes later with a spectacular free-kick from 25 yards with Jack Lester making it 4-1 in the 116th minute.
Stockport had swept ahead with their first attack after just 59 seconds. Big striker Luke Beckett, a target for United's city rivals Sheffield Wednesday, pumped in a harmless looking cross from the right flank, the ball evaded Paddy Kenny and crept in at the far post.
United had to wait until the 18th minute for their first real strike on goal, skipper Chris Morgan having a header deflected after connecting with an Alan Quinn cross and then shooting wide of the far post from ten yards.
Minutes later Jon Harley curled in a cross from the left flank as Ashley Ward connected only to pull his header wide of the target.
Stockport suffered a blow in the 28th minute, Beckett limping off after taking an earlier knock to be replaced by Chris Williams.
Williams almost snatched a second six minutes later wriggling to find space before being stopped by Morgan's block tackle.
It was the United skipper who levelled matters on 36 minutes. Midfielder Nick Montgomery sparked the moved picking out Ward, the big striker headed back into the goalmouth and Morgan steamed in to power his header past Neil Cutler from six yards.
Ward was inches away from adding United's second in the 51st minute, Barry Hayles setting up the opportunity as the ex-Bradford striker drilled the ball wide from 12 yards.
He went even closer nine minutes later leaping to meet a Leigh Bromby cross at the far post, Cutler snatching the ball on the line as the game edged towards extra time.