Adam Mace gave the visitors a 26th minute lead that Anaclet Odhiambo eventually cancelled out 8 minutes after half-time.
But Richard Thwaite restored Slimbridge`s lead with 13 minutes to go.
The turning point came in the 83rd minute when the Swans had Kieran Squires shown a straight red card and then Lewis Coyle equalised to earn the Oxfordshire hosts a point.
Keep up-to-date with our exclusive email newsletters.
Subscribe