Kane put the Sabres back in front 4-3 at 8:29, beating Mrazek from the left faceoff circle for his second goal.
"After they tied it, we talked about needing to win a 12-minute game, and we were able to do that," Bylsma said.
Lehner made 19 saves in the third period.
"We knew they were going to come after us, and our defense was doing everything they could," he said. "That was a battle."
Kane gave Buffalo a 1-0 lead at 7:12 of the first period with his seventh goal. Mrazek stopped Kane's original wraparound attempt, but he was able to flip the rebound over the goalie's shoulder.
The Sabres took a 2-0 lead at 15:54 when Reinhart scored a power-play goal. Rasmus Ristolainen took a shot from the point, and Mrazek was in position for the save, but Reinhart tipped it past him from the edge of the crease.
"It's hard to win in this League, and constantly having to play from behind makes it harder," Red Wings coach Jeff Blashill said. "You aren't going to win many when you give up four goals, either."
Zetterberg made it 2-1 at 14:46 of the second with a bad-angle shot that deflected in off Lehner.
The Sabres scored 22 seconds later to take a 3-1 lead. Eichel broke in down the left wing and flipped a shot into the top of the net.
"I had given Detroit the momentum by letting that puck go in, so it was huge when the guys bailed me right out," Lehner said.
Drew Miller made it 3-2 at 16:14, knocking a loose puck past Lehner after Dylan Larkin's shot was blocked in front.
