Michal Neuvirth made 36 saves and stopped Henrik Zetterberg in the third round of the shootout to help the Sabres (3-9-1) win in Buffalo after five straight home losses in which they were outscored 16-4.
''You hate losing more than you love winning,'' said Girgensons, who scored the winning goal. ''Winning is awesome. It's a different feeling. It feels good. I just hope we can keep it going and improve.''
Brian Flynn and Chris Stewart both scored their first goals of the season, and Tyson Strachan had two assists for Buffalo, which beat Detroit for the first time since Oct. 13, 2009.
''We had a rough October and we weathered the storm,'' Stewart said. ''We're going to learn a lot more about what this team is about in November.''
The Sabres were coming off a 5-0 loss Saturday night in Pittsburgh.
''When a team plays the night before, you should jump all over them from start to finish,'' Red Wings coach Mike Babcock said. ''We had the puck all night. We had lots of opportunities, just didn't finish.''
Pavel Datsyuk had a goal and an assist, Tomas Tatar also scored, and Jimmy Howard made 19 saves for Detroit (6-3-2), which had won two in a row before starting a stretch of three games in four nights in four cities.
The Red Wings outshot the Sabres 17-8 in the third period and 4-0 in overtime. Detroit went 0-for-4 on the power play, including one in overtime.
''It was a sloppy game,'' Tatar said. ''But we have to find our way to win these kinds of games. We had a power play in overtime. We have to score one of those.''
The Red Wings appeared to net the winning goal at 2:28 of overtime on Henrik Zetterberg's slap shot but the goal was disallowed when officials ruled Johan Franzen had established position in the crease.
''I was talking to the ref a couple times during the game,'' Neuvirth said. ''He told me, `I know he's standing in the crease, so whenever they're going to score a goal, I'm going to disallow it.' I was glad he actually did it.''
Stewart tied it 2-2 with 6:52 remaining in the third period, bouncing the puck in off Red Wings defenseman Brendan Smith off a rebound. Ennis and Strachan earned assists.
Datsyuk put Detroit ahead 2-1 early in the third period, finishing a 3-on-2 rush with Justin Abdelkader and Smith with a short wrist shot tucked inside the left post.
Datsyuk has earned a point in all six games since returning from a separated shoulder. He has five goals and five assists this season.
Detroit held the puck for nearly two minutes before Tatar's wrist shot from the left circle went through the legs of Sabres defenseman Josh Gorges and past Neuvirth to make it 1-1 with 7:05 remaining in the second period. Riley Sheahan and Datsyuk assisted on Tartar's third goal.
The Sabres scored first for just the fourth time in 13 games. Flynn gave Buffalo a 1-0 lead 6:55 into the second when he backhanded a loose puck in off the crossbar.
NOTES: Franzen returned to the lineup after missing six games with a groin injury. ... The Sabres recalled F Johan Larsson from AHL Rochester to replace the injured Marcus Foligno. Foligno got hurt in Saturday night's loss at Pittsburgh. ... Buffalo signed F Justin Bailey to a three-year, entry-level contract. Bailey, currently playing with the OHL's Kitchener Rangers, was a second-round pick in 2013. ... Detroit will play at Ottawa on Tuesday and at the New York Rangers on Wednesday. Buffalo will host Montreal on Wednesday.