The young Glovers boasted a 2-0 lead at the break courtesy of Paulius Golubickas and Nestor Shako strikes, before Winky hit back with two second-half goals of their own, forcing the tie into extra-time.
Yeovil restored their lead almost immediately through substitute Jack Rice, before Wincanton forced the lottery of a penalty shootout with a late equaliser.
The home keeper became the hero saving two spot-kicks, earning Wincanton a place in the third round at Town’s expense.
In boggy conditions at Wincanton Sports Ground, the hosts began on the front foot and forced Chandler Hallett into a save with his legs inside the opening quarter of an hour.
The Yeovil stopper almost had a moment to forget when his attempted clearance was charged down, only for the ball to trickle away from danger as the opening goal remained elusive.
However, with 25 minutes on the clock, the Under-18s took the lead after good play from captain Joe Tomlinson.
The full-back’s run into the box was eventually halted by a good challenge, but only teed up Golubickas who made no mistake from close range.
The half got better for Antonio Falanga’s side 10 minutes later, when Golubickas turned provider to slide a clever through ball for Shako to finish off.
In the driving seat at the break, Wincanton sparked the game to life by grabbing a goal back just seven minutes after the restart.
Looking threatening from set-pieces, a wide free kick was poked home to make it 2-1 teeing up a nervy end to the match.
Winky piled numbers forward in search of an equaliser and their persistence paid off with three minutes of normal time remaining.
A whipped delivery from a corner sparked a scramble in the box which eventually resulted with the ball nestling in the back of the net to force extra-time.
Looking tired, the hosts fell behind just two minutes into the first 15-minute period when Rice slotted home following good work from Neville Nzembela.
The game could’ve been put beyond doubt with 102 minutes played when Nzembela was brought down in the box, only for Golubickas to have his penalty tipped around the post.
That miss came back to haunt the Glovers as with three minutes remaining Wincanton levelled it up once again following a looped ball over the top which was coolly dispatched past Hallett.
With momentum on their side Winky netted all four of their penalties and kept out two of Town’s, bringing an end to an entertaining match which halted Yeovil’s progression in the Somerset Premier Cup.