Ajay Leitch-Smith's 24th minute goal gave the Glovers the lead at half time. The home side drew level when Joe Tumelty slotted home from close range.
The scores remained at 1-1 at the end of extra time, meaning the tie went to a penalty shoot-out.
Town won the shoot-out 4-2 after everyone converted their penalties with Matt Thorne and Jamie Lyons missing their kicks for Larkhall.
A date for the Semi Final at Bridgwater Town has yet to be set.