The Swans fell behind after Sam Gallagher's opener before Rhian Brewster's volley on the stroke of half-time and Andre Ayew's penalty minutes after the restart gave Steve Cooper's side the lead at Ewood Park.
Woodman, who had saved a penalty from Fulham striker Aleksandar Mitrović during an agonising 1-0 defeat against the Cottagers on Wednesday, kept out Danny Graham's spot-kick before Rovers midfielder Bradley Johnson denied Swansea maximum points with a 95th minute equaliser.
Elsewhere, Jacob Murphy played the full 90 minutes of Sheffield Wednesday's home clash against Derby County but the midfielder ended up on the losing side as the Owls were beaten 3-1 at Hillsborough.
The hosts conceded three goals in the opening half-hour following Tom Lawrence's brace and Jason Knight's strike, with Wednesday substitute Josh Windass netting a second half consolation.
Achraf Lazaar was an unused susbtitute during Cosenza's 1-1 draw against Venezia in Serie B while defender Kell Watts was not in action for Mansfield Town after the Stags' League Two clash against table toppers Swindon Town was postponed due to a waterlogged pitch.