Lars Eller gave the Capitals a 1-0 lead at 4:55 of the first period. After taking a cross-ice pass from T.J. Oshie, Eller scored glove side.
Colton Sceviour tied it 1-1 by tapping in a rebound off a shot from Kasperi Kapanen at 16:52 before Wilson gave Washington a 2-1 lead six seconds later. His wrist shot bounced high off DeSmith and over the goalie into the net.
Wilson then scored on a one-timer with 17 seconds remaining to make it 3-1.
"Not too worried about the individual game right now," Wilson said. "Just trying to play the system. We'll obviously take them when they go in, but we've got to get the group going where we want to be. That's the most important thing right now."
Jake Guentzel tapped in a backdoor pass from Bryan Rust on a 5-on-3, making it 3-2 with his first goal of the season at 5:45 of the second period.
"I told them we've got to have a will to win here. We've got to be a hungry team," Penguins coach Mike Sullivan said. "We didn't have it in the first period. After that, we responded. The special teams might have been the momentum swing that gave us a jolt and we were really good after that. But we can't rely on that."
After Evgeny Kuznetsov made it 4-2 at 8:16 with a wrist shot off a rebound, Blueger's goal got Pittsburgh within 4-3 at 15:07. DeSmith flipped a pass into the neutral zone, sending Blueger on a breakaway to score five-hole on a backhand.
It was the Penguins' first 5-on-3 shorthanded goal since Matt Cooke scored against the Flyers on Feb. 18, 2012.
"I was just trying to get it down the ice, and Teddy was in the right place at the right time," DeSmith said. "It was lucky, but I'll take it."
Evgeni Malkin's first point of the season, a power-play goal on a one-timer, tied it 4-4 at 17:15.
"Just can't sleep on this team," Eller said. "They score two and they're back in the game. They shouldn't have been. We should've closed the game there when we were on a 5-on-3 power play, I think. We just let them back in the game.
"The mistakes are what cost us a point today. I hope we learn from that. That's probably better than letting it slide."
NOTES: Penguins defensemen Marcus Pettersson (upper body) and Juuso Riikola (upper body) did not play the third period. Sullivan said they were being evaluated after the game. Pettersson was injured by a check from Oshie in the second period. ... Kapanen played 10:02 in his Penguins debut, which was delayed by immigration issues and a mandatory quarantine. He was acquired in an offseason trade with the Toronto Maple Leafs.
