Matthews gave Toronto a 2-1 lead at 5:24. He tried to center the puck from behind the net, but it hit Sabres forward Evan Rodrigues' skate and he got to the loose puck and scored from below the right circle.
Gauthier scored 28 seconds later to make it 3-1 on a backhand in the slot, and Ennis extended the lead to 4-1 at 13:02, taking a stretch pass from Gauthier and scoring on a breakaway.
"I don't know, we had seven or eight minutes of bad hockey where we let them get behind us and that was it," Eichel said. "We just had to stop letting them get behind us. They're a good transition team. We all know that. It's something we scouted before the game, so you have to take care of the puck and whenever they get the puck, they're looking to go. They had too many odd-man rushes in the second period and it cost us the game."
Reinhart cut it to 4-2 at 17:25 on a 5-on-3 power play, and Eichel scored nine seconds into the third period to make it 4-3.
Kapanen scored a shorthanded goal with 2:18 remaining to put the Maple Leafs ahead 5-3.