The Capitals, who beat the Edmonton Oilers on Monday, won back-to-back games for the first time this season. Pittsburgh, meanwhile has lost five in a row, the first time that's happened under coach Mike Sullivan.
Oshie, who had been knocked out of the game due to injuries in the first and third periods (stick to the face and shoulder to the face, respectively), came back in time to take a John Carlson pass and beat goalie Casey DeSmith from just outside the crease for the game-winner.
The tally snapped a 1-1 tie and gave Washington the victory in a game where the Penguins applied constant pressure. Pittsburgh outshot Washington 42-22, but Penguins goalie Braden Holtby came up big with 41 saves.
Colton Sissons notched the second hat trick of his career, Pekka Rinne had 24 saves, and Nashville beat Colorado in Denver.
Ryan Hartman also scored and Nick Bonino had three assists for the Predators, who won their fourth straight and their 11th straight regular-season game against the Avalanche. Colorado has lost four in a row, two in overtime.
Matt Calvert scored his first goal and Semyon Varlamov stopped 30 shots for the Avalanche, who lost their third straight.
Ryan Miller made 37 saves to move into a tie for 17th place on the all-time wins list as host Anaheim held off Calgary.
With his 372nd career win, Ryan matched the total of Andy Moog, who played with four teams from 1980-98. Miller also moved within two wins of tying John Vanbiesbrouck for the most NHL victories by a United States-born goalie.
Ryan Getzlaf scored the game-winner in the third period for the Ducks, who won for the second time in three games. Anaheim's Jakob Silfverberg and Adam Henrique had first-period goals. Matthew Tkachuk and Mark Jankowski scored for Calgary, which had its four-game win streak snapped.