The Royals had gone into halftime with a deserved 2-0 lead thanks to goals from Pavel Pogrebnyak and Noel Hunt.
However, Swansea pushed and pushed in the second-half and earned their reward with two goals in seven minutes.
Michu's diving header pulled one back for Swans on 71 minutes thanks to Wayne Routledge's determined wing play.
And then on 78 minutes it was Routledge who equalised, playing a one-two with Luke Moore before beating Alex McCarthy in the Royals goal.
TRIBAL ASKS: Is the manner of Swansea's comeback enough to suggest they've stopped the rot?