Jake Guentzel scored at 2:11 of overtime after Tom Wilson scored the tying goal with 15 seconds left in the third period to give Washington the point it needed to qualify.
Guentzel redirected a pass from John Marino for the game-winner and the 5-4 final.
"Heads up play by [Jared McCann] to find Johnny in the slot, and just a really nice play from Johnny there to find me at the back door," Guentzel said. "Really good play by those two and I was just on the end of it."
Wilson made it 4-4 from in front off a pass from Evgeny Kuznetsov from the corner with an extra attacker.
Kasperi Kapanen scored two goals for the Penguins (33-15-3), who have won five of six and are tied with Washington for first place in the eight-team MassMutual East Division. Tristan Jarry made 33 saves.
Pittsburgh clinched a playoff berth for a 15th straight season, the longest active streak among teams in the four major North American professional sports leagues (MLB, NFL, NBA).
"We're [going to] enjoy this one tonight," Penguins coach Mike Sullivan said. "These guys have competed extremely hard all year long to earn the privilege to compete for the Stanley Cup in the playoffs. I'm certainly proud of the group for what they've accomplished to this point. By no means are any of us satisfied. We understand that there's a lot of work to do but we're [going to] take a deep breath tonight and just enjoy it."
Dmitry Orlov had a goal and an assist, and Daniel Sprong scored his fourth goal in three games for the Capitals (32-13-5), who had won three straight. Nicklas Backstrom had three assists, and Vanecek made 30 saves.
Washington clinched a playoff berth for the seventh consecutive season and 13th time in the past 14 seasons.
"It's good. We didn't talk about it too much, to be honest with you," Capitals coach Peter Laviolette said of making the playoffs. "It was always a bigger picture than that that we were talking about. [But] you should never take anything for granted. It's a hard league, and the whole objective is to qualify to play for the Cup, so that's in the rearview mirror."
The teams play again Saturday. Pittsburgh is 5-2-0 against Washington this season, with three wins in overtime and one in a shootout.
Kapanen made it 4-3 at 10:55 of the third period when he scored on a wrist shot from the right circle on the rush.
"I think he's got such a good shot," Sullivan said. "He's a good goal-scorer. I think his speed his so dangerous. He just backs defenses off with his speed and usually that creates opportunity for him or his linemates."
T.J. Oshie gave Washington a 1-0 lead with a power-play goal at 7:27 of the first period when he deflected Justin Schultz's slap shot from the center point.
Kapanen tied it 1-1 at 15:27 when he intercepted Backstrom's pass and scored on a wrist shot from the slot.
"I don't think we gave much," Laviolette said. "The ones we gave were pretty big, so there's a couple of turnovers. They turn things around pretty quick if you don't take care of the puck and execute with it. There was a lot of good things we did tonight too."
