On the back foot for the majority of the first-half, the visitors looked to have made it to the break unscathed before Keski Anderson popped up late on.
Theo Robinson had a guilt-edge chance to make it a disastrous end to the opening 45 minutes for the green and whites but it went begging to tee up a second half with it all still to play for.
Brighter after the restart, the green and whites had a goal disallowed to deny an equaliser their second half performance deserved up until the 90th minute when Gafaiti popped up at the back post to draw the score level and put a vital point on the board.
Making the short 80 mile trip to Wiltshire, Neale Marmon made three changes for his second game in command.
After a week back in training following an ankle injury, Omar Sowunmi returned to the starting eleven to take the captain’s armband alongside Gary Warren.
Sticking with a 4-3-3 formation, Tristan Abrahams and Courtney Duffus refreshed the front line as Gabby Rogers and Alex Fisher dropped to the bench.
Making a final change to the matchday squad, 18-year-old Alex John was awarded a first senior involvement to become the fifth Academy product in 2018/19 to earn a league call-up.
Awaiting his chance on the sidelines, the game itself saw the Robins make an assured start at the County Ground that produced two early chances.
Inside the opening five minutes, Danny Rose forced Nathan Baxter into a smart save with a 20-yard volley before Robinson squandered an even better chance a quarter-of-an-hour later.
Collecting the ball just six yards out, the striker scuffed his effort to allow Bevis Mugabi the chance to hack clear as it trickled towards goal.
Maintaining their monopoly on possession, Richie Wellens’ side continued to test the resoluteness of a Town defence that was dealt a blow half-an-hour in when Sowunmi limped off on his return from injury to see Gafaiti introduced.
Producing plenty of crosses into Baxter’s box the Glovers stood firm to maintain their clean sheet as the break approached, albeit failing to test Luke McCormick at the other end.
Having seemingly ridden the storm to make it through to half-time, Swindon finally opened the scoring on 44 minutes.
Recycling possession from a wide free-kick the ball was worked to Luke Woolfenden in the box with the defender’s shot from the angle being deflected onto the crossbar and back onto the head of Anderson who couldn’t fail to score.
Causing uncertainty amongst the Town ranks the lead was almost doubled inside 60 seconds when Robinson was played in on goal, but the striker’s attempted lob dropped wide of the post as Baxter could only watch on.
Keeping it at a one-goal deficit the Glovers were quick to change things tactically after the restart as Fisher replaced Sessi D’Almeida in a change to 4-4-2.
Needing to take risks in order to get back into the game spaces began to open up at the back for Town, which Robinson almost exploited.
Beating an advanced Baxter in a foot race for the ball the striker was able to round the keeper before seeing his low shot cleared by Gary Warren inside the six-yard box.
At the other end, Daniel Ojo’s mazy run was met with a shot over the bar just past the hour-mark as the Somerset outfit began to see more of the ball.
Quickly followed by another chance, the home crowd began to vent some frustration as Abrahams’ low cross was fired into the arms of McCormick by Matt Worthington.
Cheered on by the travelling 331 supporters, Town’s good spell looked to have been rewarded with an equaliser as the game ticked into the final 20 minutes.
Tom James’ sumptuous cross was scrambled across the line by Duffus to spark celebrations that were only cut short by the linesman’s flag for offside.
Not disheartened by the decision the waves of attacks continued to come Swindon’s way as their keeper was forced into making a smart stop late on.
Substitute Ryan Seager’s neat through ball found the run of Fisher whose first-time strike had to be parried behind for a corner as the pressure continued to mount.
Ticking towards the final few minutes James almost grabbed a dramatic equaliser with a stunning 35-yard free-kick only for McCormick to spoon the dipping effort into the air and eventually away from danger.
Having pushed the play-off chasing Robins until the end, the game seemed to have gone when only two minutes of additional time were shown by the fourth official but for one last moment of magic.
James’ superb delivery from a wide free-kick was flicked on to Gafaiti at the back post who directed the ball into the bottom corner with a pinpoint header to send the away end into raptures and provide a massive boost for Town’s survival hopes.
Swindon Town: McCormick; Knoyle, Woolfenden, Conroy, Koiki; Carroll, Rose, Bennett (Broadbent 89′); Woolery, Anderson (Doughty 69′), Robinson.
Subs not used: Matthews, McCourt, Dunne, Taylor, Richards.
Yeovil Town: Baxter; Mugabi, Sowunmi (Gafaiti 30′), Warren, Ojo; James, Worthington, D’Almeida (Fisher 52′); Abrahams, Duffus (Seager 76′), Dobre.
Bookings: Duffus 20′
Subs not used: Nelson, John, Gray, Rogers.