Spurgeon and Charlie Coyle both scored within 5:32 of Dahlin's marker, knotting the score at 3-3. The Sabres came out strong in the third and regained the lead on Reinhart's goal, but Zach Parise tied it once again with Dubnyk pulled and 1:37 remaining.
"In the second period, you don't want to sound negative, but we have a 3-1 lead," Sabres coach Phil Housley said. "That's a time to really lock it down and keep a simple game. I thought we were trying to get four, but not in the right way. In the defensive zone, guys were beating us back to the net.
"Those things we've got to learn to tighten it up, but I'll give the guys credit in the third period. We found a way to win, even though we gave up that five-on-six goal, I liked our overtime and we found a way to win the shootout."
A focus throughout the season for the Sabres - and one that alternate captain Kyle Okposo referenced at practice following the loss to the Chicago - has been the importance of staying even-keeled through the ups and downs a long season.
The Sabres showed that levelheadedness on Saturday, balancing the relief of victory with the reality that they have another important game against Carolina, with whom their tied in the standings, on Thursday. They'll get back to work in the meantime.
"I mean, we can't be taking any relief out of anything right now," Reinhart said. "The schedule's so important for us. The games are kind of winding down in a hurry and we know how important every game is. We've got to play with a playoff atmosphere and take advantage of that every night.
"We have another one coming up in a couple days."