Dahlin made it 1-0 at 18:13 of the first period after his shot from the left point deflected off Lightning defenseman Ryan McDonagh's knee and went in off the post.
"We showed how good we can play," Dahlin said. "It stinks to lose but we got one point and a little bit of confidence back."
Kucherov tied it 1-1 at 11:10 of the second period. Sabres defenseman Zach Bogosian couldn't control a pass in the slot and Kucherov stole the puck and skated in with Tyler Johnson on a 2-on-1 before finishing with a one-timer.
"In the locker room after the game, you could tell how excited he was," Lightning coach Jon Cooper said. "The biggest thing for me was how excited the guys were for him and to do it in 62 (games) ... in the past 20 years, he's in some pretty elite company to be able to do that."
Sabres forward Evan Rodrigues said he missed the target on the pass that led to Kucherov's goal.
"[Bogosian] was wide open in the slot I put it 6 inches too far in front him," Rodrigues said. "If that connects it's a Grade A chance. Obviously, it bounced off his foot and went the other way. It's a tough bounce."
Steven Stamkos and Kucherov scored for Tampa Bay in the shootout.
Hutton was a late replacement for Linus Ullmark, who was scratched with a lower-body injury. He made three saves in overtime, including stopping Victor Hedman on a breakaway at 3:59.
"He really battled for us," Rodrigues said. "He was engaged. But credit to our defensemen because he was seeing a lot of pucks from the point and from the slot."
