A full-blooded affair between two in-form sides looked to be edging Bournemouth's way on the back of Harry Arter's superb 20-yard strike midway through the first half.
But when the Cherries' former England goalkeeper David James lost his footing on a sodden surface 25 yards out of his goal, Williams composed himself to slot home and earn Swindon a deserved draw.
It was an absorbing affair from start to finish, with Swindon missing two good chances either side of Arter's goal, notably James Collins, who screwed his shot wide of an unguarded goal 10 minutes before the break.
The breakneck pace of the game, played in torrential rain, continued after the break, with Arter denied by Wes Foderingham as he attempted to double Bournemouth's lead.
Bournemouth pair Brett Pitman and Lewis Grabban traded missed chances with Matt Ritchie and substitute Chris Martin before Williams equalised for Swindon five minutes from time.