The win was Calgary's first in the First Niagara Center after losing eight straight games. The Flames' previous road win versus the Sabres came in 1996 at Memorial Auditorium.
Byron has been involved in the Flames past four goals, including consecutive overtime assists, as he tries to stay with Calgary. He was recalled from Abbotsford of the American Hockey League on Nov. 28.
"Paul is such a little Energizer bunny," Calgary coach Bob Hartley said. "He works hard and he's fun to watch. Right now, he's great to put on the ice. That's two games in a row now in OT that he gives us the win.
"He's finding a way to spend Christmas in Calgary."
The teams were scoreless into the third period when Zemgus Girgensons made a strong rush down the right side of the Flames zone before sliding a backhand pass into the slot. Moulson shot it by Ramo at the 5:40 mark for his 11th goal of the season and his first in six games.
A first-round pick of the Sabres in 2012, Girgensons has five points in his past seven games.
The Flames tied the game less than 3 minutes later.
The Sabres came close to opening the scoring nearly 11 minutes into the game, when Ramo lost sight of the puck in his crease. Stajan knocked the net off its moorings before Ville Leino could score.
Calgary scored in the final minute of the first period, but the goal was disallowed and the Flames were assessed a penalty for too many men.
"One player jumped," Hartley said. "We tried to get two dollars for a looney with two players out there."
Girgensons and Tyler Myers both put shots off the post in the second period as both teams struggled to find quality scoring chances.
"I'd rather only talk about the third period," Hartley said, "Because the first 40, I don't know how it was from the press box or the stands but from the bench, but `ugly' is a gentle word."
Enroth was at his best during the second period, stopping 13 shots, but his finest moment came with a glove stop of Joe Colborne shortly after Buffalo took the lead.
"I thought he played a very good game," Sabres interim coach Ted Nolan said. "That shot at the end, it was a very quick shot. Just one of those things we have to get more than one goal."
Matt D'Agostini had a chance to put the Sabres up with under four minutes to play before shooting the puck into the glove of a prone Ramo.
The Sabres had five power plays, but their 26th-ranked power play could not produce a goal. The Flames have killed 27 of their past 28 penalties.
Calgary also struggled with the man advantage, as Buffalo killed off all four of its penalties.
Marcus Foligno took three minor penalties in the game and was benched for all but 21 seconds of the third period.
"You don't mind penalties, but you once you get lazy and start grabbing people, cross-checking people for no reason, you're better off not playing," Nolan said.
The Sabres are home again on Tuesday when they play host to Winnipeg, while Calgary continues its road trip against the New York Rangers on Sunday night.
The Sabres recalled LW Matt Ellis from their American Hockey League affiliate in Rochester before the game as Cody Hodgson missed his second game with a lower body injury ... Flames RW David Jones played his 300th career game ... Buffalo also scratched healthy defenseman Mark Pysyk and center Luke Adam.
Copyright by STATS LLC and The Associated Press
BUFFALO, NY - DECEMBER 14: Lee Stempniak #22 of the Calgary Flames controls the puck against Tyler Myers #57 and...