Former Southampton winger Dyer made an instant impact from the bench to equalise for Michael Laudrup's side, capitalising on some poor Saints defending to level the scores.
Swansea threatened to claim an unlikely victory late on with some fine attacking football but ultimately had to settle for a hard-fought point.
Dyer took to Twitter to thank Swansea supporters for making the trip to St Mary's.
He tweeted: "Great point from the boys! Dug in today and battled hard! Swansea fans were class as they always are on the road."
Laudrup's men sit in 11th place after their late comeback, with 13 points from 11 matches.