Barrie gave the Avalanche a 1-0 lead at 10:31 of the first period after getting to a loose puck and scoring on a slap shot. He has three goals in his past two games.
Parise tied it 1-1 with a power-play goal at 1:52 of the second period. It's the fourth straight game the Wild have scored on the power play (5-for-13) after going 0-for-18 in their previous six games.
Colin Wilson appeared to give Colorado a 2-1 lead at 7:49, but Wild coach Bruce Boudreau challenged for goaltender interference, and the goal was overturned after video review.
Jost took a stretch pass from Wilson and scored on a breakaway to make it 2-1 at 13:04.
"I shoot on [Dubnyk] in the summer, so I guess it was a little of a chess match," Jost said. "He's seen some of my moves and I've kind of seen some of his tendencies or whatever, so I was trying to overthink it. I was thinking five-hole, but I said no, I need to go high glove on the backhand there. It felt good to get that one."