The Sabres, who are off to a 7-0 start this season, surpassed the 11-game winning streak that began at the end of the 1974-75 season. They are one game from matching their best start ever, set in 1975.
Buffalo hasn't lost a regular-season game since April 7, against Philadelphia.
Vanek's quick shot from the slot squirted through goalie Cam Ward's legs with 6 1/2 minutes to go and gave Buffalo a 4-3 lead. Jaroslav Spacek made it 5-3 nearly two minutes later.
Jason Pominville and Dmitri Kalinin also scored for the Sabres.
Scott Walker, Eric Staal, Eric Belanger and Mike Commodore had goals for the Hurricanes, who had their three-game winning streak stopped.
Both teams traded goals in the first period before Staal gave Carolina a 2-1 lead 8:37 into the second period with the Hurricanes on a two-man advantage.
Kalinin answered for Buffalo with a short-handed goal, blasting a high slap shot over Ward's right shoulder from the left circle.
Afinogenov gave the Sabres their first lead 58 seconds later, banking a shot that went off the back of Ward's skates. Carolina tied it just over two minutes later when Belanger's shot from the side of the net ricocheted off goalie Ryan Miller's skates.
Commodore made it 5-4 with a short-handed goal with 1:15 remaining in the third.
Miller improved to 6-0 by making 16 saves, including stopping Staal on a breakaway 30 seconds into the game.
Ward, who was making his fourth consecutive start and seventh overall, finished with 24 saves.
Notes: The Hurricanes wrap up their season-high seven-game road trip on Saturday against the New York Islanders. ... Buffalo opens a three-game road trip Saturday in Boston. ... Carolina F Ray Whitney added two assists, and is one point shy of 600 for his career. ... Carolina D Bret Hedican was back in the lineup after missing two games with a hip injury. ... Kalinin's goal was Buffalo's first short-handed score this season. ... Only one penalty was called in the first period, and that came with 48 seconds remaining.