Scott Boden and Peter Winn also got in on the act as the League One visitors were sent crashing at the first hurdle by the Silkmen for the second successive season.
Defender Danny Andrew had the hosts' first sight of goal when he shot narrowly wide from distance after just six minutes, while Jennings chipped over the crossbar after being played in by Danny Whitaker.
Swindon goalkeeper Tyrell Belford was at full stretch to save a Danny Rose effort from outside the penalty area before Jennings gave his side the lead after 26 minutes following neat work from Boden.
Belford made two more impressive stops from Jennings before Swindon had their first real effort, a free-kick from Alex Pritchard which flew wide shortly before the break.
With 20 minutes remaining Boden ran in behind the Swindon defence, steadied himself and shot across Belford to make it two-nil against the run of play.
Five minutes later and Jennings put the result beyond doubt with a well-taken goal from inside the box after a nice lay-off from Andy Halls on the right.
And a fourth goal for the Silkmen arrived when Winn took advantage of a defensive error to blast home from close range in the 89th minute.