David McAllister, who spent time on loan with Shrews last term, scored the game's only goal with a fourth-minute volley from Marcus Williams' cross to give the Blades all three points.
Shrewsbury had a number of chances to score and will feel they should have secured at least a point, but that did not concern Wilson.
He said: "Overall it's a great three points at this stage of the season. Whether we deserved the three points is irrelevant because we're just delighted to come out with a win. You just want your season to get some momentum.
"It was a typical first game of the season. I thought Shrewsbury did brilliantly at times and put us on the back foot. We knew we were in for a tough game because they're a strong side who're very difficult to play against.
"There are some good teams in our division with some very effective players. We saw the work ethic we're going to need to take us through the season. We had some very clear-cut chances to put the game to bed but we defended well. There are loads of things to work on but at this stage I can't say that I'm disappointed with anything."
