Thompson took another pass from Tuch and scored on the rush 22 seconds later at 7:50 for a 4-0 lead.
He scored on the power play from the top of the left circle at 12:22 to make it 5-0, and during a 5-on-3 at 16:40 to make it 6-0.
"In 26 years pro, I've never seen that," Blue Jackets coach Brad Larsen said. "We didn't have an answer for that one line. Tage Thompson just tore us up tonight."
Laine scored at 10:49 of the second period to cut it to 6-1, and Nyquist made it 6-2 at 12:04 before Thompson's 21st of the season at 16:07 made it 7-2.
"It hasn't sunk in," Thompson said. "I don't think it will."
Thompson is the second Sabres player with five goals in a game, joining Dave Andreychuk (Feb. 6, 1986).
"That was an amazing performance by Tage, and really, the whole group set the table," Buffalo coach Don Granato said. "I thought the energy, the collective effort, focus to start was really good and enabled that to happen."
Peyton Krebs upped the lead to 8-2 at 17:19, and Tuch scored at 19:16 to make it 9-2.
Laine made it 9-3 at 4:47 of the third, and Nyquist scored at 15:49 for the 9-4 final.
NOTES: Thompson has 37 points (20 goals, 17 assists) in his past 19 games. He has 14 points (eight goals, six assists) in a five-game point streak. … Thompson is the second United States-born player to score five goals in a game, joining Mark Pavelich (New York Rangers, Feb. 23, 1983). … Thompson is the 18th NHL player with four goals in one period, and the first since Patrick Marleau (San Jose Sharks, Jan. 23, 2017). … Skinner has 10 points (three goals, seven assists) on a five-game point streak. … Cozens has a five-game point streak (12 points; five goals, seven assists). … Dahlin has a goal and eight assists during a five-game point streak. … Blue Jackets forward Johnny Gaudreau had an assist to give him 10 points (two goals, eight assists) in a six-game point streak. He has 19 points (four goals, 15 assists) in his past 12 games.