Jennings opened the scoring when his turn and shot found the bottom corner.
Scott Boden doubled the lead when he latched onto a through ball and slotted into the corner before Jennings's super close-range finish made it three.
Peter Winn's point-blank strike rounded off a convincing win for the non-league side who move into round two.
Victory for Macclesfield, over a side 55 places above them on the football ladder, was their seventh in the last nine games and maintains their unbeaten record against Swindon in the competition.
Nick Chadwick's goal was enough for Conference Premier side Tamworth to upset League Two Cheltenham in a minor upset.
But elsewhere, everything went much to the formbook. A hattrick from Paul Gallagher helped Preston to a 6-0 romp over non-league Barnet, both sides finishing with ten apiece.
The only non-league success against league opposition came when Newport were held to a 1-1 draw by Braintree and Boreham Wood held Carlisle United to a goaless draw.
Those looking fo a major upset will be hoping Bishop's Stortford at home to Northampton in Sunday's televised game or Shortwood United at home to Port Vale in Monday's live TV game can conjure up some Cup magic.