Washington gave up a two-goal lead in the period before Wilson redirected a centering pass from John Carlson on the power play to make it 3-2 after Pittsburgh defenseman Kris Letang's stick broke.
"When a guy loses a stick it's another advantage in our favor," Wilson said. "We were able to execute and it was in the back of the net. That's a fortunate break for us, but at the end of the day we've still got to put it in, and the power play did a good job to do that."
Nicklas Backstrom, Lars Eller and T.J. Oshie each had a goal and an assist for the Capitals (10-5-4), who split the two-game set. Carlson had two assists, and Vitek Vanecek made 26 saves.
Brandon Tanev and Jake Guentzel scored, and Tristan Jarry made 30 saves for the Penguins (10-7-1), who had won three straight.
"I loved our resilience," Pittsburgh coach Mike Sullivan said. "There was a lot of fight in getting back in the game. We come back and get those two goals and I thought we had pretty good momentum. So it was unfortunate that we didn't find a way to at least get a point out of this, but that's hockey."
The Penguins have won four of the six games between the teams this season, including two in overtime and one in the shootout. Pittsburgh won 3-2 in overtime here Tuesday.
"We walked away with a point [Tuesday] but tonight … more reflects what we want to see and what we are trying to get to," Capitals coach Peter Laviolette said. "So it was nice to follow up [Tuesday] with a win. I think we did things the right way and we got rewarded for it."
Backstrom gave Washington a 1-0 lead at 11:20 of the first period when he worked a give-and-go with Nick Jensen.
Oshie made it 2-0 at 3:18 of the third period when he slid the puck past Penguins defenseman Mike Matheson on the rush and then dove to poke it in for his third goal in the past three games.
"I mean what an individual effort to make that happen," Laviolette said. "He's been good. I think that he's been a consistent player for us the entire year, and it is nice to see him get rewarded. To me that was a lot of individual effort and a really nice play to finish it off."