Buffalo and the Capitals have 88 points in the Eastern Conference, though Washington won the tie-breaker in the race for the final playoff spot because they have more non-shootout wins. Both have two games remaining.
"Close isn't good enough. We want to get into the playoffs," Buffalo's Jordan Leopold said.
The Sabres trailed 3-0 after the first period on Tuesday and were down 5-3 with less than six minutes to go but fought back bravely.
Alexander Sulzer netted a goal, his second of the game, and Jordan Leopold tied the contest with less than two minutes remaining in regulation.
Derek Roy then delivered the game-winner at 3:29 in overtime, also his second score of the night, and the home team celebrated their biggest victory of the season.
"I just wanted to step up and do my part," Roy said. "I told myself before the game that my part is to score goals."
The battle for the Western Conference's final post-season spot heated up as San Jose beat Dallas 5-2 to move three points in front of the ninth-place Stars.
The Sharks are in eighth one point behind Phoenix, who have 93 after shutting out Columbus 2-0 to stay in front.
With the season winding down, the New York Rangers put the finishing touches on their top seed in the East with a 5-3 triumph over the Philadelphia Flyers.
"A great feeling knowing we finished first," goalie Henrik Lundqvist said.
"But when we play the next game it doesn't mean anything. Yeah, we have home-ice advantage, that's a big thing, but after that it doesn't really get us anything."