Larsson lifted the rebound of Miller's shot that bounced off the end boards to give the Sabres a 1-0 lead at 1:32 of the second period.
Conor Sheary made it 2-0 when he redirected Miller's point shot at 9:42.
Korshkov capitalized on a Buffalo turnover with a one-timer from the right face-off circle to cut it to 2-1 at 11:29.
Zach Hyman tied it 2-2 when he deflected Jake Muzzin's point shot at 2:11 of the third period.
Jack Eichel put the Sabres ahead 3-2 with a power-play goal from the left circle at 6:06.
Kyle Okposo made it 4-2 off a Maple Leafs turnover in their own end at 6:55.
"As a team we did really well," said Okposo, whose goal gave him his 500th NHL point. "I thought we handled the mostly ups we had in the game. They made a push there at the beginning of the third, they tie the game, [Hutton] bails us out, makes a couple of huge saves, and then we're able to roll from there. So I really like the way that we played the full 60 tonight."
Jimmy Vesey extended the Buffalo lead to 5-2 at 7:37.
"We just haven't been able to put a full 60 together," Maple Leafs captain John Tavares said. "At times we haven't held leads very well, and tonight's a night we just didn't play very good overall. ... But considering the sense of urgency we need to have and the points, how crucial they are, it's disappointing, especially once we gave ourselves an opportunity in the third to win it, to not find a way and raise our game."