Striker Barry Corr was given a straight red for an off-the-ball incident in the 20th minute, but Swindon chased and harried and covered to deny the visitors until the 87th minute.
Thomas Butler then scored what he must have thought was the winner, but Swindon poured forward to gain a dramatic late penalty.
Earlier Swansea had shown the more attacking intentions, but neither goal was seriously threatened until Guillem Bauza reacted well to lob the ball past Petr Brezovan. Miguel Comminges reacted excellently to get back and head the ball off the line.
Swansea continued with most of the possession but could not pierce a hard-working home defence and when Darren Pratley finally managed a 25-yard shot, he found Brezovan perfectly positioned as the half came to an end.
Swansea continued to dominate after the break but, despite their possession, failed to find any cutting edge as the visitors were restricted to long-range efforts, mainly off-target.
Gary Monk thought he had won it with a 74th minute header, but Brezovan made an excellent reflex save to tip the ball past the post.
Butler finally broke the deadlock following an Alamo-type siege in the Swindon box, firing home from 16 yards.
The game then erupted in the final minutes. Hasney Aljofree had a header cleared off the line, Comminges hit the post as Swindon surged forward and the visitors' defence looked a little on the creaky side.
Referee Melin then turned down a Simon Cox appeal for a penalty as it was Swansea's turn for their box to be under siege.
With four minutes stoppage time allowed it was almost complete when Kristian O'Leary was adjudged to have fouled Cox.
Three bookings for dissent, a delay of three minutes and Cox stepped eventually up to equalise from the spot and give the home side a deserved point.