The Blades were unbeaten in the league at Bramall Lane and facing a team who had not won in their last 22 matches, but veteran striker Steve Howard headed Pools in front after seven minutes.
Michael Doyle equalised courtesy of a mistake by goalkeeper Scott Flinders before lifelong Sheffield Wednesday fan Ritchie Humphreys restored the visitors' lead early in the second half and Howard headed them a two-goal cushion after 72 minutes.
Harry Maguire's stoppage-time strike was too little, too late for the Blades, who began the day top of the table but slipped to second place after their surprise defeat.
"Overall, we can't really complain too much about the result because they took their chances better than we did and we defended poorly in key areas at key times in the game and that gave us a mountain to climb," said Wilson.
"You've got to roll your socks up and have a go and in fairness to Hartlepool, they did that. We had the chances in the game and that's the biggest disappointment. We haven't constructed goals when we had good chances.
"We had a very dodgy offside decision against us and we had a penalty turned down when Harry Maguire was brought down right at the end. If we had got one of those then it might have been a different game but unfortunately we didn't."
Wilson added: "We missed Shaun Miller and Neill Collins greatly. Shaun has been a threat in recent games and Neill is a key defender for us.
"We also had Tony McMahon out. That created a big void in the team, with all due respect to the players who came in."