Surviving an early scare when former loanee Connor Roberts hit the post, Town enjoyed the better of the first half and took the lead on 25 minutes.
Carl Dickinson’s corner was neatly flicked in to allow Jordan Green to net from just a few yards out as the hosts impressed.
Growing in confidence to dominate the remainder of the half, 10 outfield changes at the break saw the Glovers lose their stranglehold on the match and eventually concede the equaliser with 13 minutes remaining.
Yan Dhanda, off the bench, reeled away in celebration after his strike beat Trialist B before Matty Grimes won it in stoppage time with an unstoppable strike.
The first of six matches in preparation for the 2018/19 campaign, summer signings Dickinson and Diallang Jaiyesimi ran out in Green & White for the first time.
Fellow acquisition Gary Warren started on the bench, whilst loan goalkeeper Nathan Baxter remains with parent club Chelsea until next week due to Thibaut Courtois and Willy Caballero’s World Cup commitments.
Darren Way cast the eye over seven unnamed trialists as well as released defender Dan Alfei who’s been invited back for pre-season, but was unable to call on the injured duo of Jake Gray and James Bailey.
Bathed in Somerset sunshine, the Swans looked to make their two-tier advantage count and almost took the lead inside the opening five minutes.
Wayne Routledge found Roberts in the box, with the full-back’s strike coming back off the post after the intervention of Omar Sowunmi.
Having survived the early scare Way’s side went on the offensive themselves and tested visiting stopper Erwin Mulder just before the 10-minute mark.
Green’s strong running won a free-kick wide on the right and from the resulting set-piece the flying winger forced the Dutch keeper into a save from his edge of the box shot.
In the ascendancy, the Glovers enjoyed a flurry of corners midway through the half which resulted in the deadlock being broken.
Dickinson’s low delivery was flicked on by Alex Fisher, falling perfectly into the path of Green who slid to convert from close range.
More than matching their Championship opponents Way’s side remained on the offensive and Mulder was called into action again 10 minutes before the break, this time to clear Dickinson’s drilled cross with his feet.
Becoming one-sided, Green continued to be a thorn in the Swans’ side and almost doubled his tally after brilliantly beating two visiting defenders only for his curling effort to float just wide.
Home keeper Stuart Nelson remained untested at the break as Town’s first 45 minutes since May showed promise.
In true pre-season tradition, both sides made copious changes at the halfway stage; the Glovers changed all 10 outfield players whilst the Swans made six changes themselves.
Disturbing the flow of the game, the visitors became dominant after the restart enjoying a monopoly on possession.
A couple of long-distance strikes stung the hands of Nelson before the veteran was replaced for the final half-an-hour by Trialist B as the match lost some of its earlier quality.
Remaining on top of proceedings Graham Potter’s side continued to press for an equaliser and almost found one after strong running from Dan James.
The full-back’s cut-back was blocked in the box before a scramble resulted with Yeovil hacking clear the danger.
As time continued to tick away it took until 77 minutes for the Jack Army to make their dominance count, as substitute Dhanda leveled it up with a deflected volley from inside the box.
Looking for a win in his first game in charge, Potter’s men continued to press for a winner and found one in stoppage time as Grimes’ strike from range left Trialist B with no chance to condemn Town to defeat in a game of two halves.
Yeovil Town: Nelson (Trialist B 60′); James (Alfei 46′), Sowunmi (Warren 46′), Donnellan (Mugabi 46′), Dickinson (Trialist E 46′); Green (Trialist C 46′), Trialist A (Trialist D 46′), Santos (Spencer-Clarke 46′), Browne (Trialist F 46′); Jaiyesimi (Trialist G 46′), Fisher (Zoko 46′ (Ojo 89′)).
Subs not used:
Swansea City: Mulder; Roberts (Dhanda 66′), Amat (Fernandez 46′), Van Der Horn (Roden 66′), Harries (Fulton 46′), Naughton (Lewis 66′); Carroll (Maric 46′), Byers (James 46′), Routledge (A. Ayew 46′); Narsingh (Blair 66′), McBurnie (Grimes 46′).
Subs not used: Zabret, J. Ayew.