Girgensons was Buffalo's eighth shooter in the tiebreaker, after the Sabres blew a two-goal lead in the third period.
"We could have gone home 10 minutes quicker, but it was great to get the win," Girgensons said.
The two squads, which met in the 2006 Eastern Conference finals before the Hurricanes won their only Stanley Cup, came into the game as the only NHL teams without a point.
Buffalo led 3-1 early in the third period, but Chris Terry scored for Carolina (0-2-1) and then assisted on Zach Boychuk's goal that tied it at 18:12.
Foligno had his first two goals of the season, and Torrey Mitchell also scored for the Sabres (1-3).
Cody Hodgson, moved from left wing to center after recording only one shot on net in the first three games, had Buffalo's other goal in the shootout.
"When you're going through what we're going through, you have to take baby steps," second-year coach Ted Nolan said. "I've never questioned the character of this team one time. We weren't playing very well for whatever reason, but we have to find a solution to our problems and tonight we had some big performances. Overall, sometimes you'll take an ugly win over a beautiful loss."
Riley Nash and Terry each had a goal and an assist for Carolina. Terry was the only Hurricanes player to score in the shootout.
"The effort was good," first-year Carolina coach Bill Peters said. "You're not going to get many points in this league down two with 10 (minutes) to go. I liked our third period. I thought we could have been better in certain areas, but we went 2 for 2 on penalty killing, so that's a positive."
The Hurricanes played without captain Eric Staal (upper body) and wing Patrick Dwyer (lower body), both injured Saturday against the New York Islanders. Carolina has also been without Jordan Staal (broken leg) and Jeff Skinner (concussion) all season.
All but Jordan Staal, out at least three months, are now listed as day to day.
Carolina defenseman and former Sabres player Andrej Sekera left with an upper-body injury in the first period and did not return.
The Sabres, who had the league's worst record last season, led 1-0 after the first. They outshot Carolina 13-7 and had two power-play opportunities to none for the Hurricanes in the period.
Buffalo scored at 12:07 of the first, when Foligno beat Ward stick side off Chris Stewart's pass across the crease.
Enroth stopped ex-Sabre Nathan Gerbe on a short-handed breakaway in the last minute of the period.
The Hurricanes scored on a rush 30 seconds into the second, when Nash put in a rebound of Jay Harrison's shot. But the Sabres answered at 1:53 when Mitchell scored from the left circle.
Foligno made it 3-1 at 2:54 of the third, beating Ward stick side from the slot.
"Everybody's got confidence now," Foligno said. "This win is really going to help out and we can focus on the next game. We had a lot of support around the puck, especially in the first two periods. And when we were at the line we had a lot of good, smart dumps."
Terry scored for Carolina off a faceoff at 12:11.
Buffalo had just two shots in the third period.
NOTES: Buffalo had lost four straight at PNC Arena. ... The Hurricanes' next game is Thursday night at the New York Rangers. Buffalo plays at Florida on Friday night. ... Carolina alternate captain Harrison skated in his 300th game for the franchise. ... Cody Hodgson was moved from left wing to center for Buffalo after recording just one shot on goal in the first three games. ... Buffalo G Matt Hackett (lower body), RW Patrick Kaleta (upper body), D Jake McCabe (middle body) and D Mark Pysyk (upper) each missed a fourth straight game. ... D Brett Bellemore was Carolina's only healthy scratch, while C Brian Flynn, D Andrej Meszaros and D Nikita Zadorov were all healthy scratches for the Sabres.