Dustin Byfulgien tapped in a perfect pass from Evander Kane midway through the third period for the game-winner in a game that had a playoff atmosphere.
The loss snaps Washington's four-game winning streak and allows the Jets to move to within two points of the Capitals with 11 games remaining.
"Buff (Byufulgien) likes to get out there and make a difference, and he was," Winnipeg coach Claude Noel said.
"He was resting in the crease, recovering in the crease. Evander Kane made a heck of a play and fed Buff there, and what can you say?
"Buff wants that impact in games and does. It was good. It was really good."
Andrew Ladd opened the scoring for Winnipeg in the first period but the lead lasted just 22 seconds as Brooks Laich tipped in and equalizer.
Nikolai Antropov put the Jets ahead again late in the second period, outmuscling Alex Ovechkin for position in front of the Capitals net to backhand a shot past Tomas Vokoun.
However, the Jets squandered a chance to add to their lead when two Capitals penalties in quick succession gave Winnipeg a two-man advantage before the end of the period.
Washington, by contrast, took full advantage of a powerplay of their own when they made it 2-2 on Mathieu Perreault's shot through the legs of Ondrej Pavelec in the opening minute of the third period.
A penalty to Winnipeg defenseman Ron Hainsey with just over two minutes remaining created some anxious moments as Washington pulled their goalie in favor of an extra attacker but the visitors were unable to find a third tying goal.
"This one stings a little bit more than a playoff loss," Washington's Troy Brouwer said. "Because a playoff loss, you forget about it and move on.
"But this one, we really needed the points in the standings to try to get some separation between us and Winnipeg. But now it's going to make this race a little bit tighter."