Jenner pushed it to 4-1 at 19:42, making a move around Owen Power in the left circle before lifting a backhand over Anderson short side.
Quinn brought the Sabres to within 4-2 at 5:02 of the third period, and JJ Peterka cut it to 4-3 at 9:46 when he scored with a one-timer from the right circle on a power play.
It was Peterka's first goal since Dec. 13 (27 games).
"Obviously, we know the magnitude of this game and the magnitude of the moment," Buffalo captain Kyle Okposo said. "I just thought we were playing pond hockey in the first period, and it's not a good recipe for success this time of the year. Third period we came out and we pushed, 'Quinny' scores a big goal, get a big a big one on the PP there, and we've got 10 minutes to tie it up and we weren't able to find the last one."
Robinson completed his hat trick with an empty-net goal and 19:25 for the 5-3 final.
NOTES:Prior to the game, Columbus traded injured forward Gustav Nyquist to the Minnesota Wild for a fifth-round pick in the 2023 NHL Draft. … Goalie Joonas Korpisalo and defenseman Vladislav Gavrikov were scratched for "trade-related reasons." … Thompson has 80 points (41 goals, 39 assists) in 59 games, the fewest needed by a Sabres player to reach the mark since 1992-93 (Pat LaFontaine, 42 games; Alexander Mogilny, 47 games) . … Thompson (six goals, five assists) and Skinner (five goals, five assists) each extended his point streak to seven games.