The Penguins overcame a late goal in regulation by the Capitals to reduce their deficit in the best-of-seven Eastern Conference semifinals to 2-1. Pittsburgh will try to even the series at home Friday.
Pittsburgh also halted Washington's club-record playoff winning streak at five games. The Capitals came back against the New York Rangers after falling behind 3-1 in that series and then grabbed the first two games of their showdown with the Penguins.
The duel continued between Washington's Alex Ovechkin and Pittsburgh's Sidney Crosby. Ovechkin scored his eighth goal of the playoffs to go with an assist and Crosby recorded two helpers.
Crosby assisted in setting up Letang's game-wining blast.
Ovechkin scored just 83 seconds into the game, but the Penguins came back to take the lead on Ruslan Fedotenko's second-period goal and a power-play score by Evgeni Malkin with 4:49 to play in the third period.
Washington forced overtime, however, when Nicklas Backstrom scored on the power play with 1:50 to go in the third.
Marc-Andre Fleury needed to make just 21 saves to get the win. Varlamov gave up three goals on 42 shots.