Olofsson took a pass from defenseman Henri Jokiharju in the low slot to give the Sabres a 1-0 lead 53 seconds into the first period.
Zacha backhanded a rebound on the power play to tie it 1-1 at 2:06 and scored at 3:47 from the top of the left circle on the power play for a 2-1 lead. Hughes extended the lead to 3-1 at 16:04.
Thompson made it 3-2 on the power play at 11:56 of the second period. Skinner tied it 3-3 when he backhanded a loose puck at the right side of the net at 18:21.
"I think that that's just a sign of us having some more confidence that we've generated over the past week and a half or so," Sabres forward Kyle Okposo said. "Just making little, subtle plays, 'Tommer' gets a big one on the power play there. 'Skinny' gets one in front to tie the game. But you got to try to use that, 3-3 game, you don't have a good 40 minutes and you've got a chance to win a game, we've just got to come out and have a better third, and we just didn't have it tonight."
Sharangovich made it 5-3 from the right circle at 18:17, and Kuokkanen scored an empty-net goal at 19:28 for the 6-3 final.
"Really liked our third period where we generated some good opportunities," Devils coach Lindy Ruff said. "We didn't give up a lot. We only gave up a couple of shots. We played the game the right way, put pucks in deep. We got rewarded. The Boqvist goal was get the puck to the net, it went off the goaltender's pad, and came right back out and 'Bo' put it in the back of the net."