Jennings' two-run homer in the seventh inning off reliever James Curtis (2-4) gave the Catfish (13-9) a 4-3 lead. He also had an RBI bunt single in the third.
Joey Callender ripped a pair of doubles and scored twice and Ryan Royster collected three hits and a walk for Columbus.
Reliever Matt Falk (3-0) allowed a run on three hits, striking out two without a walk in three innings for the win. Ryan Reid pitched around three walks and fanned two in two hitless frames for his seventh save.
The win leaves the Catfish in 3rd place, 1-1/2 games behind Augusta and Asheville.