Ross McCormack struck a 16th-minute winner for the visitors with a neat finish from Matt Smith's knock-down for his 29th goal of the season.
Barnsley, who won 2-1 at fellow strugglers Charlton earlier in the week, never threatened to stage a comeback.
Defeat leaves the Tykes three points from safety with three games left to play.
"It wasn't a good result, obviously, and it doesn't make it any easier, but who said it was going to be easy anyway?" Wilson said.
"We thought it was going to go right to the wire and I think I'm right. We didn't offer enough quality and I thought the game was poor. I thought it was a typical derby game without the goalmouth action.
"There were not a lot of chances and there was one bit of quality. Of all the people to fall to, it was McCormack and you didn't want it falling to him with the form that he's in at the moment. That was the difference.
"We huffed and puffed and they huffed and puffed a bit. We had good situations developing that we didn't take advantage of. It was a frustrating afternoon.
"We're ready for the challenge - we have to be. There are three big games to go and we're very confident that we can go to Derby and Middlesbrough and get results.
"If we don't believe we can do it, we might as well go home now."
Leeds manager Brian McDermott felt that victory for his side was the right result.
"I think it was well deserved," he said.
"I think the game was separated by a bit of real quality. Ross' first touch was fabulous and then a great finish.
"He's a match winner and those players are very, very important. He took the one chance that made the difference. He's right up there as a finisher.
"After that, I thought we deserved to win the game. They were tough conditions - it was really windy and the pitch was a bit bobbly at times.
"We have kept two clean sheets in two games and we're delighted with that.
"There are a lot of things that have gone on in the last three months. We needed the stability and the ownership is sorted out now.
"I never worry about my own future. I'm doing the best I possibly can and I've done the best I possibly can in the last year that I've been at Leeds.
"I might look back on this last three months and think that I've done well to dig in there because I've certainly had to do that.
"There's definitely a different air around the place and we can look forward now, which is the important thing."