The 23-year-old made 116 as Tim Groenewald and Lewis Gregory rattled through the rest of the hosts line-up.
Jennings, who hit 15 fours, finally fell top-edging Josh Davey to gully before Durham were bowled out for 256.
Tom Abell and Davey fell lbw to Rushworth, then Onions bowled Marcus Trescothick as Somerset closed on 30-3.
Durham batsmen Mark Stoneman and Scott Borthwick both fell cheaply as Somerset started brightly before Jack Burnham (33) helped repair the innings in a 79-run third-wicket partnership with Jennings.
Burnham eventually fell when he was bowled by Davey.
Gregory then claimed two wicket in two balls, including uprooting Paul Collingwood's middle stump first ball, as Somerset looked to gain some momentum.
Rushworth became Gregory's fourth victim of the day, departing to an exceptional slip catch from Jim Allenby it left the visitors with 12 overs to face.
And last year's Division One leading wicket-taker struck in his fourth when Abell was trapped playing across the line before nightwatchman Davey was caught out in the next over when the ball nipped back.
Onions' inswinging yorker then dismissed the Somerset captain as the away side limped to the close, trailing by 226 runs.
Keaton Jennings' century and three quick wickets from opening pair Chris Rushworth and Graham Onions gave Durham the edge against Somerset on day one.