Laich and Jason Chimera worked a 2-on-1 toward the Ottawa net in the final seconds of the extra frame. Chimera skated in on the left side and fed a crossing pass to Laich, who beat Craig Anderson for the game-winner.
Troy Brouwer and Nicklas Backstrom scored in regulation for the Capitals, which broke a four-game losing streak and logged their first win under new Coach Dale Hunter.
Michal Neuvirth turned aside 32-of-34 shots to earn the win.
Anderson finished with 32 saves.
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