With the Sabres on the power play, Vanek picked up a loose puck in front and slid it past Florida goalie Alex Auld to give Buffalo its second straight win and 13th overall.
Brian Campbell, Paul Gaustad and Ales Kotalik also scored for the Sabres, who have come back to win six times after falling behind by two goals this season.
Auld stopped Vanek on a penalty-shot attempt at 1:37 of the second period.
Gary Roberts, Ruslan Salei, Joe Nieuwendyk, and Steve Montador scored for the Panthers, who lost their second straight.
The Sabres played for the first time without three regulars, including starting goaltender Ryan Miller, who is day-to-day with a lower body muscle strain. Forward Maxim Afinogenov (upper body) and defenceman Henrik Tallinder (broken left forearm) also missed the game.
Martin Biron started in place of Miller and made 15 saves in his second straight start, and fourth of the season. He won a franchise-record 13 consecutive starts last year when Miller was out for more than a month with a broken thumb.
The Panthers opened the scoring when Gary Roberts scored his third goal on Florida's first shot five minutes into the game.
Salei made it 2-0 11 seconds after Vanek missed on the penalty shot. Coming out of the penalty box, Salei took Jay Bouwmeester's long pass and wristed a shot past Biron for his second goal.
Vanek solved Auld for his ninth goal to make it 2-1 over a minute later, and Campbell tied it 2:13 later with his second goal.
Gaustad's first goal gave Buffalo its first lead at 3:14 of the third, and Kotalik made it 4-2 50 seconds later with his fifth goal.
Nieuwendyk closed the gap to 4-3 midway through the third with his fourth, and Montador tied it with 3:21 left in regulation.
Auld made 21 saves in losing his sixth straight start.
Florida dropped to 1-6-2 on the road.
Notes: The Sabres look to extend their league-record eight-game road winning streak to start the season Saturday in Philadelphia. . . . Florida has now lost seven of its last nine. . . . Panthers centre Chris Gratton missed the game to be with his wife, who gave birth Thursday to the couple's second child. . . . Auld received an assist on Salei's goal. . . . Briere's assist on Campbell's goal gave him points in 13 of Buffalo's 15 games. . . . Both teams finished with just four shots on goal in the first period.