Visiting goalkeeper Lewis Clayton gifted the hosts the lead, fumbling Joe Tomlinson’s cross into his own net before second-half strikes from Tom Handley and George Lloyd turned the match on its head.
The first period brought little action on goal with Glovers stopper Charlie Philpott called upon just once, making a smart save to his left.
Yeovil began to threaten just after the half hour mark and went ahead thanks to a strike of luck.
Tomlinson’s inswinging delivery from the right looked to be an easy claim for Clayton but the Robins number one misjudged the flight of the ball and fumbled the ball over his own line.
Antonio Falanga’s outfit failed to get out of second gear after the interval and were pegged back on 75 minutes.
The home side couldn’t clear their lines from a corner and Handley reacted to level.
Things got worse for the junior Green and Whites with a minute left when the referee pointed to the spot.
Lloyd stepped up and converted from 12 yards to complete Cheltenham’s turnaround and send the Gloucestershire club through to the semi-finals.