Lars Eller and T.J. Oshie also scored for the Capitals (11-12-4), who are 2-2-1 on their season-long six-game road trip. Charlie Lindgren made 28 saves.
"I thought every guy in there played really hard," Washington coach Peter Laviolette said. "Came out the right way and did the right things in the first period. We had gotten away from that for a couple of games. I thought we did a good job tonight of trying to push it and trying to control the game. Tonight was really complete."
Connor McDavid scored in his fifth straight game, and Skinner made 47 saves for the Oilers (14-12-0), who won four of their previous five. Leon Draisaitl's five-game goal streak ended.
"We weren't as quick and physical as we wanted to be in the defensive zone," Edmonton coach Jay Woodcroft said. "Our goalie stood tall. We're 2-2 going into the third period. We made a critical error, and it ended up in the back of our net."
Skinner made 22 saves during a scoreless first period.
"I think if you get a few [early] chances on you and make all the saves, it's a little bit of a confidence booster," Skinner said. "They got on the power play and I got a few shots on the power play, so after that I was ready to go."