Ryan Miller made 31 saves for the Northeast Division-leading Sabres, who have won two straight. Tyler Myers and Paul Gaustad both recorded two assists for Buffalo.
Byron Bitz scored the lone goal for the Panthers.
Tomas Vokoun stopped 17 shots for Florida before he was replaced by Scott Clemmensen in the second period. Vokoun was also pulled from Tuesday night's game against Washington, a 7-3 Florida loss. Clemmensen finished Saturday's game with 16 saves.
The Sabres went ahead 3-1 on Grier's goal 13:49 into the third. Gaustad dug the puck out of a scrum in front of the net and Grier pushed it in.
Florida had closed the gap to 2-1 with Bitz's goal at 11:04 of the second. Bitz, who returned to the lineup Saturday after missing four games because of a lower body injury, tipped in the puck from in front for his fifth of the season.
Kaleta chased Vokoun with his goal 2:54 into the second. Vokoun came out wide of the net to reach a Buffalo shot, which then bounced off the end boards and came back out in front. Kaleta put in the rebounding puck with a wrist shot to stretch Buffalo's lead to 2-0.
Buffalo started the scoring on Hecht's goal with 6:34 left in the first. Hecht fired a shot from the goal line, and the puck hit Vokoun before trickling into the net.
The Sabres and the Panthers will both play again Sunday. Florida will face in-state rival Tampa Bay at home. Buffalo will visit Carolina.
NOTES: This was the third of four contests between the Panthers and the Sabres this season. They next face each other March 31 in Buffalo. ... Before the game, the Panthers unveiled the South Florida Hockey Den of Honor, which showcases important figures and moments in team history. Former Florida players, including Scott Mellanby and Brian Skrudland, attended the ceremony.