With the Panthers playing on short rest, the Sabres were the aggressors early, firing off the first four shots of the game and taking the lead on Hodgson's goal 2:06 into the first period.
Zadorov had the secondary assist on Hodgson's goal, giving him his first career multi-point game. Pat Kaleta set up the goal with a backhand pass from the boards. It was Kaleta's first point of the season and his first assist since March 7, 2012.
Foligno scored to make it 2-0 with 6:55 left in the first period. It was the sixth time in seven games the Panthers have fallen behind in the first period.
"We have to be better in the first period," Florida defenseman Brian Campbell said.
Campbell brushed off the suggestion that playing on the second night of a back-to-back contributed to the Panthers' poor start.
"I don't think good teams use that excuse," he said.
Florida was better in the second period, scoring on its first two shots to tie the game less than five minutes in.
Bergenheim knocked the puck in off Zadorov's skate 1:08 into the period, then sent a pass across the crease to set up Hayes tying goal at the 4:51 mark.
Zadorov was on the ice for both of the Panthers' second-period goals.
"That was just a bad couple of shifts, bad couple bounces, bad couple goals," Zadorov said. "I just forgot about that, stepped on the ice and did my job."
Jokinen gave Florida its first lead 57 seconds into the third period, scoring on a slap shot from the right circle off a crossing pass from Dave Bolland.
Gionta took a long pass from Tyler Myers and scored on a wrist shot from the top of the right circle to tie the game 3-3 with 14:33 left in the third period.
NOTES: The Sabres were down to four healthy defensemen by the end of regulation. Andrej Meszaros left the game in the second period after he was hit in the face by a deflected shot. Josh Gorges, Buffalo's alternate captain, was injured while blocking a shot in the third period. Nolan said both players will miss Monday's game against Ottawa. . Sabres D Mike Weber underwent surgery on his left knee Friday and will miss about two weeks, coach Ted Nolan said. . Florida has gone to overtime in 11 of its 28 games, tying Anaheim for the most in the league.