Jacob Mellis stunned the league leaders by firing fourth-bottom Bury into a 12th-minute lead, only for Billy Sharp to level nine minutes later with his 17th of the season.
Caolan Lavery’s low cross went in off Kelvin Etuhu on 72 minutes before the recalled United striker squared for Kieron Freeman to tap in his second in two games to seal United’s sixth straight win.
Wilder said: “Bury had a right go at it, we knew what type of game it’d be.
“We had to really work hard and show a bit of quality at the end to get our noses in front. The pitch wasn’t great, that made it more competitive, but we showed again there are different ways to win a game.
“They got a bit of a leg-up. Their goal came from possibly Caolan Lavery’s one mistake of the afternoon, and it was some strike. No keeper’s saving that.
“It was quite an open, end-to-end game, but I don’t think it was any surprise when we got in front.
“A couple of teams around us haven’t won games they were expected to win recently. We were expected to win here and we’ve managed to get a big three points.”
Bury: Williams, Soares, Kay, Bedeau, Leigh, Mackreth (Miller 79), Etuhu, Mellis (Danns 65), Burgess, Styles (Pope 79), Vaughan. Unused substitutes: Cooney, Dudley, Hope, Rachubka.
Sheffield United: Moore, Ebanks-Landell, Wright, O’Connell, Freeman, Basham, Fleck, Lafferty (Coutts 79), Scougall (Duffy 54), Lavery, Sharp (Done 90). Unused substitutes: Brown, McNulty, Wilson, Ramsdale.
Referee: R Lewis (Shrewsbury).