Bjugstad scored on the rebound of a Carson Soucy shot to give the Wild a 2-1 lead at 3:42.
"They fought their way through a game that we're fortunate to get two points in," Minnesota coach Dean Evason said. "We got better and better as the game went on. We thought our third period was real good. Our energy was good in the third period. We put a lot of pressure on them, played a lot in their zone."
Cam Talbot made 24 saves, and Ryan Suter scored his first goal of the season for the Wild (19-10-1), who lost their previous two games.
"I think we're going through a bit of a slump with energy, but we're finding ways to win, which is important," Suter said. "Every game can't be [where] you score six goals and everybody is smiling. Every once in a while, you've got to grind them out like we did tonight."
Troy Terry scored for the Ducks (9-18-6), who have lost six of seven. Ryan Miller made 21 saves.
The teams will play again here Wednesday.
"It's hard not to give Talbot credit," Terry said. "He played a really good game. I don't know, I thought we created a lot. There's a lot to hang our heads on this year that we need to learn from. I don't necessarily think that game was one of them. I think we can learn from it."
The Ducks outshot the Wild 10-5 in the first period and finally broke through when Terry scored with a backhand on a partial breakaway at 5:17 of the second for a 1-0 lead.
But Suter tied it 1-1 at 16:21 of the second with a shot from the point through traffic for a power-play goal.
"Big goal by [Suter]," Bjugstad said. "That was a great all-around play on the power play. Obviously we've had our chances, [but] to come through with a goal like that after being down one, that's game-changing."
