The Blades had collected just two points from their previous five games, a run which had seen them slide from League One leaders down to sixth place.
But it all came right for Wilson's men on their first visit to the Greenhous Meadow as two goals inside nine second-half minutes from captain Michael Doyle and Dave Kitson, a cracking 25-yard strike, proved decisive.
Shrewsbury, now just four points clear of the relegation zone, pulled a late goal back as Stephen McGinn's long-range free-kick seemed to elude everyone on its way in, but they were unable to prevent a first home defeat since October.
"It's a massive win for us," said Wilson. "This game was important for both clubs and maybe on the selfish side it was more important for us because we had to maintain that important push for the top.
"The expectation level is very high at United so it was a great win for us."
Wilson felt his side were worthy winners although they had to be patient having been denied by a sting of fine saves from Shrewsbury goalkeeper Chris Weale on his 31st birthday.
Wilson added: "It was a good team performance and there was a great work ethic. I felt we were in control of the game for large periods.
"Possession-wise we always looked in control and we looked very confident on the ball.
"More chances arrived in the second half and we got a blinding goal from Kits and the other one was more workmanlike when we pressured the ball and got the rewards at the end of it."
- Sports & Recreation
- Dave Kitson