Laich deflected the game-winner at the 1:00 mark of overtime to cap a two-goal comeback for the Caps.
Keith Aucion and Mike Green scored in the third period for visiting Washington. The period began with New York up 2-0 on Radek Martinek's short-handed goal in the first period and Jeff Tambellini's backhander at 3:44 of the third period.
Green cut the deficit to 2-1 at 6:02 of the third period with a shot from above the right circle. Aucion tied the score with 6:52 to play on a wraparound shot that beat Dwayne Roloson.
Roloson had 21 saves for New York. Washington's Jose Theodore stopped 28 shots.