Ryan Miller made 35 saves for the Sabres, who snapped a three-game losing skid. Matt Moulson, Brian Flynn and Steve Ott also scored for Buffalo in the shootout.
With his 46th shootout win, Miller took sole possession of the record for most shootout wins in NHL history. The Rangers' Henrik Lundqvist has 45.
Milan Michalek scored Ottawa's goal and Robin Lehner made 35 saves. The Senators' shootout scorers were Mika Zibanejad, Kyle Turris and Michalek.
Buffalo is 4-0 in games decided by a shootout.
Jamie McBain appeared to give the Sabres a second-period lead at the 15:30 mark when Lehner pushed a loose puck into his own net, but referee Tom Kowal ruled that the whistle had blown.
The play was reviewed and fans were treated to a replay with the on-ice volume amplified that showed the puck crossing the line before the whistle blew.
The NHL released a statement saying there was a delayed penalty on Ottawa which should've stopped play when Ottawa touched the puck.
The Sabres hadn't played since Saturday and controlled much of the first two periods against a Senators team playing a third game in four nights.
Ottawa scored first on a 2-on-1 rush 6:25 into the first period, when Michalek beat Miller for his fifth goal of the season.
Buffalo nearly tied it when Erik Karlsson took a hooking call at the 9:14 mark of the second period, but Lehner made four saves on Buffalo's first power play of the night to protect the 1-0 lead.
Girgensons had the game-tying goal.
Christian Ehrhoff took a wrist shot that Lehner kicked to his right and Girgensons swatted at the loose puck. It beat Lehner between the legs, hitting each skate before crawling over the goal line.
The Sabres had a season-high 18 shots in the second period, and their 29 shots through two periods were more than they've had in 21 full games.
Ottawa controlled the third period as it outshot Buffalo 13-6.
Overtime had free-flowing play with scoring chances for both teams. Miller made a sprawling stop on Chris Phillips before McBain was unable to convert two chances in the Ottawa end.
Tyler Myers was led off the ice in overtime after taking a Jason Spezza shot to the face.
Ehrhoff nearly put the Sabres in front early in the third on a breakaway attempt that fooled Lehner, but went wide of the goal.
NOTES: Sabres LW Ville Leino was a healthy scratch for Buffalo. "He didn't practice all that well," Sabres coach Ted Nolan said of Leino, who is goalless in 19 games. "I think we're part of our environment. If we don't practice hard, how are you going to play hard?" ... D Eric Gryba and C Mike Hoffman were scratched for Ottawa. ... Mike Weber was again scratched for the Sabres.