The result leaves MK Dons six points adrift of safety with six games left to play, having failed to win in five matches.
Wolves had won their only previous league game at Milton Keynes when they took 9,000 fans to Stadium mk in League One in March 2014, but they fell behind when Hause turned in under pressure from Nicky Maynard.
The visitors had winger Mychal Zyro taken off on a stretcher just before half-time, but they responded well to earn their first away league win in seven attempts.
Dons keeper Cody Cropper got a hand to Saville’s header but could not keep the ball out, before Price’s first goal since March 2015 earned Kenny Jackett’s side all three points.
MK Dons manager Karl Robinson:
“We weren’t good enough, that was a very poor 60 minutes of football – after the first half-hour I thought we stopped pressing properly, our mind-set was poor, we looked nervous and we’ve got nobody to blame but ourselves.
“Today I have no protection and I have to accept the punishment that comes our way.
“For some unknown reason, we came out in the second half and we stopped playing, we just stopped passing the ball and we looked to eradicate pressure by kicking the ball long.
“My biggest disappointment is that some people aren’t learning lessons at this level. I’m not going to say who or where – we’re asking people to do certain things and we’re not learning our lesson.
“There’s only so long I can persevere with that and I might be too late.”
Wolves manager Kenny Jackett:
“It was a very good response in the second half.
“It didn’t feel in the first half we could get in behind MK Dons, we couldn’t turn them.
“We had good possession up until two-thirds of the way up the pitch then we lost our way, we couldn’t get in behind them and they broke off our bad play.
“It was a sloppy goal we gave away, but we responded well and the second-half performance was very good.”