The Glovers twice came from behind to come away from the County Ground with a share of the spoils in Sky Bet League Two.
Starting brightly, the Robins were 1-0 up early on when the experienced head of Matt Taylor popped up in the box to lash home following a dominant start from Phil Brown's side.
In an otherwise tame first period, Lewis Wing went close to equalising with a venomous strike from range, to leave a second half with plenty to play for.
Keen to not suffer back-to-back league defeats, the Glovers rallied after the break and on 68 minutes levelled through a moment of brilliance from Lewis Wing who rifled home from long range.
Before many of the travelling faithful had stopped celebrating, the Robins retook the lead in fortuitous fashion when Taylor's deflected strike earnt him a second goal of the night.
Leaving only a little over 10 minutes left to play, Town rallied for a second equaliser and sent the away end into raptures when Alex Fisher's superb volley on 89 minutes secured a point and continued Yeovil's excellent run on the road.
After naming an unchanged side for three consecutive matches, Darren Way made four changes for the short trip to Wiltshire.
Losing Ryan Dickson and Francois Zoko to suspension, Bevis Mugabi and Otis Khan replaced the duo whilst Alefe Santos and Oscar Gobern came in for Jordan Green and Jared Bird in a formation change to 4-3-3.
Bolstering options on the bench, Corey Whelan and Connor Smith returned to the substitutes bench along with Ryan Seager who sat out the weekend’s loss to Luton.
In an altered shape, the Glovers were slow out of the blocks as the hosts started on the front foot at the County Ground.
Youngster Scott Twine, making his home debut, headed just wide early on before a spell of pressure brought the opening goal after 11 minutes.
Kaiyne Woolery pulled out wide to the left, working the space to drill a low cross into Taylor who neatly dispatched at the near post.
Looking most threatening from getting the ball out wide, Swindon's best chance to grab a second came from an almost identical template to their first.
This time Taylor turned provider to cross from the right, with Twine unable to find the target with his volley from just a few yards out.
Plateauing out into a largely unnoteworthy first half - not helped by a lengthy stoppage as Town captain Stuart Nelson received treatment for a wrist injury - neither goalkeeper was called into action.
With the break approaching, Way's side registered their best chance of the half on 44 minutes when neat build up led to Wing's rasping strike from 25 yards being pushed over the bar by home stopper Stuart Moore.
Ending the first period brightly, Yeovil were dealt a blow during the break as Nelson succumbed to his earlier injury. Town's captain was replaced by Jonny Maddison between the sticks with Green also replacing Santos in a tactical switch.
Straight into the action, Maddison kept the score at 1-0 five minutes after the restart, making himself big to smother Twine's effort from the angle.
The 23-year-old's saved looked to have been an important one when 10 minutes later Sowunmi met Khan's wide free-kick, but the defender's header fell agonisingly wide of the post with Moore scrambling as the Glovers rallied.
However, the Somerset outfit didn't have to wait long to level the score and it came in stunning fashion.
Opening up 30 yards from goal, Wing let fly with an unstoppable strike which nestled in the far corner to mark his third in Green & White.
Looking like the team in the ascendancy, the Glovers were dealt a sucker punch just four minutes after equalising.
Taylor was once again the menacing figure, with his strike from the edge of the box deflecting into the back of the net past the full-stretch of a hapless Maddison.
Back to square one Town upped the tempo in the final 10 minutes and once again found a way back into the match in the dying embers of the game.
Sowunmi's flick on was majestically brought down by Fisher who volleyed in off the post to grab his fourth in as many games to earn Way's side a noteworthy point at play-off chasing Swindon.
Swindon Town: Moore; Knoyle, Purkiss, Lancashire (McGivern 63'), Hussey; Twine (Elsnik 88'), Dunne, Banks, Taylor; Richards, Woolery (Mullin 80').
Subs not used: Anderson, Henry, Linganzi, McDermott.
Yeovil Town: Nelson (Maddison 46'); Mugabi, Sowunmi, Donnellan, James; Wing, Santos (Green 46'), Gobern; Browne (Barnes 73'), Fisher, Khan.
Bookings: Sowunmi 42', Gobern 55'
Subs not used: Whelan, C. Smith, Wing, Seager.