The 19-year-old was starting just his second match for the Glovers since joining on loan from Premier League side Tottenham Hotspur in January.
Harrison almost gave Yeovil the lead just before half-time but was thwarted by Grecians keeper Christy Pym.
Strike partner Francois Zoko did put Town ahead in the 62nd minute, collecting Matty Dolan’s slide-rule pass and slotting into the far corner before Harrison got in on the act.
The London-born forward, who was denied by Pym again moments before his goal, was in the right place at the right time to nod home after Ben Whitfield’s venomous shot was parried by the Exeter stopper.
Harrison was substituted in the 77th minute to rapturous applause from the 835 Green and Whites who had made the short trip across the border into Devon.
He watched on as Alex Lacey’s thumping header put Town three-up before Exeter scored three times in the closing stages to salvage an unlikely point.
Despite the disappointing result, it was a day to remember for Harrison who took to Instagram to express his delight:
Watch Harrison’s finish on YTFC’s official YouTube channel.