The Capitals reached 86 points for the season, tying them with idle Buffalo for the eighth and final playoff spot in the Eastern Conference. Buffalo, however, has a game in hand on Washington.
Both the Sabres and Capitals are two points behind Ottawa.
Boston and Washington played two periods of scoreless hockey before the offenses took over.
Dennis Wideman and Marcus Johansson scored 2 minutes apart during the first half of the third period to give Washington a 2-0 lead.
Both teams produced two goals during the first three shootout rounds, but Laich beat Tim Thomas high to the glove side for the winning score.
Washington survived an early goaltending change to get the win. Tomas Vokoun returned from a six-game absence brought about by a groin injury, but he aggravated the injury and Michal Neuvirth came on as an emergency replacement.
Neuvirth stopped 19-of-21 shots to get the win. Thomas made 21 saves.
Boston would have clinched a playoff spot with a victory. It needs only one more point to do so.