Ollie Norwood scored a first-half penalty shortly before the Blades had striker Gary Madine was sent off, but they stood firm and substitute David McGoldrick added a second late on.
Wilder said: “Regardless of what happens this season, from my point of view, what the players have produced tonight goes a long way in terms of what I wanted when I came to the football club.
“They gave everything and from a character point of view and the art of defending, that was right up there.
“We’ve got players who are disciplined and organised and want to keep the ball out of the back of the net.
“We rode our luck a couple of times and the keeper’s made some wonderful saves when he’s needed to.
“It was the toughest of tests with the players they’ve got and their movement.”
“I thought it was a red card straight and Gaz is devastated. He’s had a lunge at the lad, the referee has seen it and I think it was the correct decision.”
The opening goal came in the 26th minute when Norwood sent Luke Daniels the wrong way from the spot after Yoann Barbet brought down George Baldock inside the area.
But United were reduced to 10 men just nine minutes later when Madine was shown a red card following a late challenge on Ezri Konsa.
United goalkeeper Dean Henderson had to be at full stretch to keep out an effort from Neal Maupay and Kamohelo Mokotjo hammered a shot against Henderson’s left-hand post
But McGoldrick sealed the victory in the 84th minute, heading in from close range after John Fleck’s corner was headed back by Jack O’Connell.