The Blades battled back from a goal down to stretch the advantage at the top of Sky Bet League One.
Billy Sharp scored his 17th of the season and Kieron Freeman is second in three days, either side of an own goal.
Wilder made changes to his starting line-up for the first fixture of 2017. Wright, Scougall and Lavery all started. Coutts and Duffy dropped to the bench after a busy Christmas period and Clarke missed out after picking up a knock on Saturday. McNulty was recalled from his loan at Bradford City and was on the bench.
The only opportunity of the opening 10 minutes was a Blades break, started by Moore and involving Lafferty and Scougall before Sharp sliced wide. Bury had to wait until 12 minutes before their first chance but they opened the scoring - Lavery was dispossessed in his own half and MELLIS was afforded the time and space to fire into the top corner, giving Moore little chance.
United, however, were level inside 10 minutes. Following a foul on Lavery, Fleck whipped in a free-kick and after a minor melee, the ball dropped for SHARP to fire home with his left foot.
The leveller prompted a spell of Blades pressure. A weighted O'Connell pass presented an opportunity for Lavery, who turned inside Bedeau but goalkeeper Williams made a fine save. The lively Lavery then was involved in two moves that saw shots from Scougall and Fleck blocked.
Late on in the half, Sharp robbed Kay but was unable to find the target with another left footer, whilst on the stroke of half-time Burgess hit a free-kick into the Blades wall after Basham's foul.
There was a lively start to the second half. Lavery and Sharp linked well but were denied by an offside flag and Bury had a great chance from Mackreth's cross but Vaughan hammered wide on 50 minutes.
Ebanks-Landell headed over a Fleck free-kick and Basham shot across goal and wide ahead of Bury thinking they should have had a penalty when a shot from Burgess hit Wright's hand but the referee was not convinced.
It got worse for Bury with United's next serious attack. Freeman cut out a pass and raced into the box, laying off for Lavery to cross and ETUHU diverted the ball past his own goalkeeper.
After a number of substitutions, including Bury throwing on a couple of strikers, United added a third on 81 minutes. Very patient build up around the left saw Lavery progress around the back of his marker and deliver a low cross for FREEMAN to sneak in and poke home from close range.
Bury hit back. A Burgess free-kick was deflected wide and on 88 minutes a misplaced Coutts pass put Vaughan in the clear but the striker's shot came back out after hitting the underside of the crossbar.
However, after the above incident, United were professional and disciplined in the closing stages and saw out the four minutes of added time well to open 2017 with a victory in front of 3,157 travelling Unitedites.
Bury: Williams, Leigh, Etuhu, Kay, Vaughan, Mellis (Danns 65), Soares, Burgess, Bedeau, Styles (Pope 79), Mackreth (Miller 79).
Unused: Dudley, Rachubka, Hope, Cooney.
Blades: Moore, Fleck, O'Connell, Basham, Scougall (Duffy 54), Lavery, Sharp (Done 90), Wright, Freeman, Ebanks-Landell, Lafferty (Coutts 79).
Unused: McNulty, Ramsdale, Brown, Wilson.