WATCH: All Wild vs. Penguins highlights
He backhanded a shot past goalie Marc-Andre Fleury to give Minnesota a 3-2 lead 2:56 into the third. Staal, who has 51 points in 50 games against the Penguins after the goal and two assists Thursday, scored one goal in his previous seven games.
"I've played them a lot," Staal said. "Been out east for a long time (primarily with the Carolina Hurricanes) and played these guys on many occasions. They've been on the right side of it a lot, and fortunately for me, it's a night where we got the better of them, but they're a good team. They have a great team every year. It's a challenge that I'm always excited to be a part of, and fortunately tonight I was on the right side of it."
Jason Pominville scored an empty-net goal to make it 4-2 with 30 seconds left.
Pittsburgh (9-3-2), which was outshot 44-41, lost in regulation for the first time in eight home games this season.
"It is a concern," Sullivan said. "I think it's hard to win consistently in this league if you're giving up 40-plus shots a game. Having said that, we got 40-plus shots in the game as well. But I don't think that's the type of game that we're trying to play."