The Capitals (52-15-12), having already clinched the Presidents' Cup for securing the best points total in the league, won their third successive game and extended their franchise-best win total.
They won 50 games in the 1985-86 and 2008-2009 seasons.
"It's a big deal (winning the Presidents' Trophy) but there's plenty of hockey left," Washington goaltender Jose Theodore told reporters. "You've got to turn the page and there's not too much time left to celebrate."
After falling behind 1-0 on a snap shot from Washington center Nicklas Backstrom, Boston took the lead on goals from Dennis Wideman and Patrice Bergeron before the Capitals' Mike Knuble tied it 2-2 with 29 seconds left in the second period.
Both teams were scoreless in the third where Theodore made 12 of his 28 saves to help force overtime.
Laich's game-winner was set up when Wideman was called for high-sticking, giving the man-advantage to the Capitals, and he capitalized by beating rookie goaltender Tuukka Rask.
"It's a tough, tough way to lose a game like that but we'll take the point," said Rask, who finished with 27 saves.
"It's always better to get (two points) but I thought we played good today. I thought we kept them outside pretty good, you don't have to give them much because they can score."
Alex Ovechkin recorded two assists but was held without a goal for a fourth successive game.
The two-time Hart Memorial Trophy winner for the league's Most Valuable Player, trails Pittsburgh Penguins center Sidney Crosby by one goal for the league lead and is two points behind Vancouver's Henrik Sedin for the most points.
Boston have three games remaining, including one more against Washington, to secure their playoff berth and are just one point behind Montreal for sixth spot in the Eastern Conference.
(Editing by Greg Stutchbury)