Bastianini made the most of a mistake by Jerel Ifil to score from six yards.
But Yeovil were still left looking for their first league win of the season as Swindon fought back to win.
Swindon were level after 24 minutes when Gareth Whalley's low cross was diverted at full stretch into his own net by Yeovil captain Terry Skiverton.
Bastianini almost restored Yeovil's lead just before the break but Swindon goalkeeper Tom Heaton dived to his left to make an outstanding save.
Swindon took the lead in the 50th minute when Jack Smith hit the bar with a shot and Rory Fallon headed in the rebound.
Colin Heath put Swindon further ahead in the 66th minute when he took a pass from Ricky Shakes to score with a fierce right-foot shoot.
Yeovil wasted a great chance to pull a goal back in the 73rd minute when new loan signing Andrew Gurney fouled Nathan Jones to concede a penalty.
Lee Johnson struck the spot-kick low and hard to Heaton's left but the goalkeeper got down to save.
Yeovil had defender Efe Sodje sent off in the 86th minute for an incident with Fallon but Yeovil were still not finished.
With three minutes remaining Skiverton scored at the right end when he headed in a corner from Jones to set up a fascinating climax.
Deep into injury time Heaton made another important block to keep out a left-foot shot from Bastianini and the ball was cleared to the other end where substitute Christian Roberts raced away to add a flattering fourth.