Enda Stevens celebrated his first call-up to the Republic of Ireland squad with the opening goal after 29 minutes.
David Brooks doubled the Blades’ lead after 64 minutes, just a couple of minutes after coming on as a substitute, producing an impressive finish following a fine interchange between Billy Sharp and Mark Duffy.
“It was a tough game,” said Wilder. “It was important that we got the first goal, but they were always in the game.
“We struggled to break them down and find the killer pass. Credit to the opposition because they were stubborn.
“They’ll take points off people between now and the end of the season and I wouldn’t be surprised if they get enough points to stay up because of the pressure on other clubs around them.
“Fortunately for us, we took our chances and won the game. I thought we started the second half really well and upped the tempo a bit.
“We never really felt out of sight until the last 15-20 minutes.
“We knew it was a big week for us. The pressure is on us if we want to be involved at the end of the season and we need to recognise that.
“We’ll not change our approach from what it has been all season.”
Lloyd Dyer’s early shot forced United keeper Jamal Blackman to palm the ball wide.
The opening goal came when George Baldock found Stevens inside the area and the wing-back finished with ease.
Dyer was put through on goal again and his fierce shot forced a great save from Blackman.
Burton appealed for a penalty when Marvin Sordell went down under a challenge inside the area and Darren Bent headed wide.
Brooks made it 2-0 before Lee Evans clipped the top of the bar late on and Stephen Bywater made a fine save to keep out John Fleck’s free-kick.