Boro, who had captain Grant Leadbitter sent-off midway through the first half, saw Daniel Ayala reduce the arrears early in the second half.
Johnny Howson went close to equalising in the latter stages, putting a curling effort just inches wide.
But Wilder said: “When you look at the quality of the opposition and the magnitude of the game as well, we played really well.
“We’ve played well recently and not got a win. I thought we were outstanding in the first half.
“Credit to the opposition, but I thought we contributed to our own downfall in the second half with the goal we conceded. We got tight and a little edgy.
“We have to learn from that. We need to be a bit more composed and calm when games become tight. They had nothing to lose and went for it.
“It was important that we got three points to keep the season going and we’ve done that.
“The defeat against Barnsley was damaging but we’re in the game and I’m expecting an electric atmosphere on Saturday when we face Millwall.”
Middlesbrough boss Tony Pulis was unhappy with the red card decision.
He said: “For the second booking, if there was contact it was minimal - knowing that another booking takes the kid out of the game.
“In this country, it’s a game of challenges and referees have got to understand there are going to be times when players make honest challenges and they are going to give free-kicks away.
“In such a big game, to make that decision was very, very poor and very disappointing.
“I thought in the first half we never played with the tempo and the energy that you need to have. The three subs came on, gave us loads of energy and I thought in the second half we were absolutely fantastic.
“For me, if you can’t finish in the top two, you want to get in the play-offs, so this is where you want to be. This is not pressure, this is what we all want.”