Kevin Doyle's 70th minute header settled a frenetic encounter which had seen Wolves twice throw away the lead. A Stephen McManus own goal had given them the initial advantage only for Grant Leadbitter to put Middlesbrough back on level terms in controversial circumstances.
Bjorn Sigurdarson regained the initiative for Wolves with Leadbitter then striking again before Doyle - match winner in the previous match against Bristol City - sealed a third victory in four games.
Wolves are now just a point from safety and Saunders said: "The game was unbelievable to watch for a neutral. Both teams could have scored ten. We went for it. We had to win and you are standing on the side there hoping we can take the chances.
"If we keep winning I think the way other results are going we have a chance of getting in the play-offs. Everyone has got momentum going now and we had to win. If we can win again at Birmingham City on Monday we could be just four points away from 10th spot."
Middlesbrough have won just one of their last nine games and have taken only a point from their last nine matches on the road.
"I stopped looking at the table in my mind a few weeks ago," said manager Tony Mowbray. "We just have to win football matches. If we win games let's see where the season takes us. But obviously we are finding it difficult to matches at the moment.
"Yet if we had win at Derby County on New Year's Day we could have gone top of the table and that is the stark reality of our position. Confidence does get eroded with defeats and yet the defeats probably come along because you don't have key players available.
"I felt that we gave Wolves a tough game but the reality is that we did not get any points. It is about getting points and I could not see us losing the game.
"But if you analyse our season we have conceded way too many goals for a team that can get out of this league. That needs to be addressed but in the meantime we have to attack games as draws are no good for us at the moment."
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- Grant Leadbitter
- Kevin Doyle