Tyler Cooke, Dave Broglino, Kevin Lawrence, Adam Toy and Ben Packman all got on the scoresheet in the South West Peninsula League West clash.
Despite the scoreline, the visitors did put in a good fight and hit Penryn's woodwork three times.
But Athletic's back four put in a fine display throughout the 90 minutes as they dealt with any ball crossed into their penalty area.
Cooke opened the scoring in the tenth minute when he slid onto Russell May's low cross into the six yard box at the near post.
It would prove to be the only goal of the half which saw Connor Menton and Liam Ames both sent shots crashing onto the crossbar for Town.
Borough were able to create some breathing space in the 55th minute from the penalty spot after Rob Stapleton was judged to have handballed in e area, much to the central defender's disagreement.
Broglino made no mistake from the spot-kick and in the 77th minute Cooke put through Lawrence who then jinked around the goalkeeper to tap in, although Town protested he had been in an offside position.
Stapleton nearly made amends three minutes from time for the penalty decision when he headed the ball onto the post, but Borough still found time for two goals in extra time.
The first came from Packman who met Broglino's cross at the back post before Toy smashed home a thunderous strike from 25 yards straight into the bottom right hand corner of the net.
Athletic's next game comes on Friday evening away at Penzance.