Zach Parise and Travis Zajac each had a goal and an assist for the Devils, who have won four straight road games and four of five overall.
Langenbrunner snapped a 2-all tie 16 seconds into a tripping penalty on Brian Pothier, chipping a shot from between the circles with 9:56 left in regulation. Washington had tied the game less than a minute earlier on Alex Ovechkin's 35th goal.
Brooks Laich also scored for the Capitals, who have lost six of seven and were bidding for a home-and-home weekend sweep of the Devils.
Instead, Clemmensen - in his second start of the year - made 23 saves for his first victory since March 24, 2006. New Jersey averted its first losing streak since dropping three straight from Dec. 12-16. Martin Brodeur had started the Devils' previous 32 games.
Laich put Washington up 1-0 16:02 into the opening period. John Erskine's drive from the left point careened off the skate of New Jersey's Jay Pandolfo in the left circle and Laic knocked in the rebound from the right side.
New Jersey tied it on Parise's team-high 26th goal 5:34 into the second. Stationed between the circles, Parise converted a left-wing feed from Travis Zajac.
Zajac was at the right post to give the Devils a 2-1 lead with 18:23 remaining in the third period. Parise's shot went between Johnson's pads, bounced off his right skate and Zajac popped in the loose puck at the left post.
But the lead lasted only until Ovechkin got his first goal in six games with 10:47 to go. Using defenceman Paul Martin as a screen during a dash down the left wing, Ovechkin snapped a wrist shot from the left circle.
Notes: Parise has five goals in seven games. ... The Devils scratched RW Brian Gionta (groin) for a sixth straight game, while Washington RW Eric Fehr (sore back) sat out his second consecutive game. ... Attendance was announced at 14,585, but a winter storm kept many seats empty.