After two scoreless periods, the Capitals appeared on their way to a win when Alex Ovechkin and Alexander Semin scored in the third.
Both those goals came on the power play, but Washington then took two penalties that cost them a win.
Evander Kane scored a power-play goal for the Jets with 2:15 to go in regulation and Dustin Byfuglien had another just 12 seconds later.
Ovechkin opened the shootout with a goal, but Ondrej Pavelec stopped the other two Washington attempts, while Wheeler and Little were producing the deciding scores.
Pavelec made 32 saves in the win. Tomas Vokoun stopped 25 shots for the Capitals, who despite earning a point fell back into a tie for the Southeast Division lead with Florida.
Wisconsin business offering 'therapeutic cuddling' forced to close
Puzzle-maker slips 'Murdoch Is Evil' into Rupert Murdoch's Sunday Telegraph