Stapleton got the best of Michal Neuvirth with a shot from the left circle to bring Winnipeg an unlikely win. The Jets trailed by three early in the second period.
Ben Maxwell and Bryan Little scored in the second period to cut Winnipeg's deficit to one and Spencer Machacek tied it with 3:45 left in regulation. Maxwell and Machacek both produced their first goal of the season after recently being called up from the American Hockey League.
The win boosted Winnipeg to within four points of the final playoff spot in the Eastern Conference. Ondrej Pavelec stopped 20 shots in the victory.
Alex Ovechkin scored two of Washington's three early goals but Washington lost for the fourth time in five games. The Capitals are tied with Buffalo for eighth place in the East.