With the game tied at 2, Jenner put a wrist shot past James Reimer to put Columbus up for the first time off assists from Sean Monahan and Kirill Marchenko.
Jet Greaves stopped 39 shots for Columbus.
Rasmus Dahlin and John-Jason Peterka each had goals for the Sabres. Reimer finished with 27 saves.
Takeaways
Sabres: Buffalo, which will miss the playoffs for the 14th straight year, has lost only twice in its last 10 games.
Blue Jackets: Columbus staved off elimination from the playoff hunt and moved within six points of Montreal for the second wild-card spot in the Eastern Conference.
Key moment
Columbus went on the power play for the first — and only — time in the game when Buffalo's Sam Lafferty was called for slashing Zach Werenski at 14:37 of the third period. Jenner took advantage with the go-ahead score.
Key stat
While Columbus went 1 for 1 on the power play, Buffalo went 0 for 1.
Up next
Sabres: Play at Florida on Saturday night.
Blue Jackets: Host Washington on Saturday.
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AP NHL: https://apnews.com/NHL