Zach Werenski increased the lead to 4-2 at 8:16, and Gaunce made it 5-2 off a pass from Laine at 17:43.
Laine, who had two assists, has scored 18 points (10 goals, eight assists) during the streak, his second-longest in the NHL behind a 15-game point streak in 2017-18 with the Winnipeg Jets.
Kyle Okposo cut it to 5-3 at 19:28 when video review confirmed the puck completely crossed the line. He said the Sabres can't use fatigue as an excuse for the loss.
"We've played back-to-backs before. We just didn't get off to a good start," he said.
Gustav Nyquist scored an empty-net goal at 19:24 of the third period to push the lead to 6-3, and Justin Danforth scored at 19:56 for the 7-3 final.
"Whenever you get contributions up and down the lineup it's huge, it's infectious," Blue Jackets coach Brad Larsen said.