The Swans made two changes from the side beaten at Port Vale with Roberto Martinez and Andy Robinson coming in for the injured Kris O'Leary and for Leon Knight
It only took four minutes for Swansea to open up the home defence when a neat pass from Owain Tudur-Jones gave Robinson a shooting chance but Rhys Evans came out quickly to block.
Robinson should have opened the scoring from a Lee Trundle corner in the 11th minute, but his header from point blank range soared high over the bar.
It was just before the half hour when Willy Gueret had to make his first save of the match when he went down low to collect a shot from Chris McPhee.
On the stroke of half-time Trundle sent a drive spinning beyond the far post with Evans beaten and Alan Tate was booked for protesting when a goal-kick was awarded.
Swansea made a double substitution on the hour with former Swindon striker Rory Fallon and Adrian Forbes replacing Adebayo Akinfenwa and Leon Britton.
But the breakthrough came at the other end one minute later when Ricky Shakes was first to a Keith Lowe back header and side-footed into the net to give Swindon the lead totally against the run of play.
Robinson finally put his side on level terms when he crashed a shot into the corner of the net in the 72nd minute from a pass by Trundle.
Once again it was all Swansea and they deservedly went ahead six minutes later when Forbes beat the offside trap and lashed the ball home with an angled shot.