After watching his shot deflect off the post and wide of the net 7 minutes into the third period, Green got a much better bounce on another shot from the same spot with 40 seconds remaining.
Green's goal -- which struck the left post and banked in off the back of Sabres goalie Jhonas Enroth's arm -- capped the Capitals' two-goal, third-period rally. Matt Hendricks and Alex Ovechkin then scored on Washington's two shootout attempts to seal a 4-3 win on Saturday night.
"No, I didn't call 'bank.' It was more of a wish than anything," Green said, describing his goal. "I came back to the bench and said, 'I wish it would have hit him in the back and went in.' And the second one did, so somebody was listening."
Green added an assist, while Ovechkin and Troy Brouwer also scored in regulation in helping the Capitals (16-17-1) improve to 4-1 in their last five. With the win, Washington took over sole possession of 11th place, a point ahead of Buffalo, in the Eastern Conference standings.
Braden Holtby stopped 20 shots through overtime, and didn't allow a goal in the shootout.
"These are the type of games that we are going to need to steal to make a playoff push," Green said. "A big win for us."
It was an all too familiar finish for the inconsistent Sabres (13-16-6).
Buffalo is 0-1-2 in its past three. The Sabres have squandered two-goal leads six times this season, and they were coming off a game in which they blew a third-period lead in a 5-4 shootout loss at Florida on Thursday.
"It's happened too many times," forward Ville Leino said. "It shouldn't happen. Good teams usually win those games."
Leino scored twice, while Christian Ehrhoff also scored for Buffalo on a day the team already began looking to the future.
Hours before the game, Buffalo traded veteran defenseman Jordan Leopold to St. Louis in exchange for two draft picks -- including a second-rounder. Leopold was in the final year of his contract and Sabres general manager Darcy Regier said he's focusing on improving the team beyond this season.
The Sabres jumped to a 3-1 lead despite being outshot 27-15 after 40 minutes.
The Capitals got a lift while killing off a delay-of-game penalty to John Carlson early in the third period.
From behind his net, Holtby swept the puck up the left boards. The puck dribbled past Sabres defenseman Andrej Sekera at the point, which set up Brouwer to score for a 2-on-1 break.
Buffalo has allowed a league-worst six short-handed goals.
Green's goal fueled the Capitals' momentum entering overtime.
Mike Ribeiro nearly sealed it for Washington 2:46 into overtime, but was robbed by Enroth's diving save.
Enroth, who stopped 35 shots, then gave up goals on both shootout attempts.
Hendricks scored by faking to his right and then going to his left, where he flipped a shot over goalie's pad.
After Holtby got his right pad out to stop Buffalo's Tyler Ennis, Ovechkin sealed the victory by snapping a shot through Enroth's legs.
"It's a credit to our guys for sticking with it," Holtby said. "Greenie came up with a huge goal, and that's what the leaders on your team do."
Ovechkin extended his point streak to seven games, in which he has seven goals and 11 points. He has 32 points (20 goals, 12 assists) in 30 career games against Buffalo. ... Sabres leading scorer Thomas Vanek (upper body) missed his second straight game. ... Ehrhoff played his 600th career game.