Gary Megson's side were condemned to their sixth game without a win in League One as fine strikes from Adrian Forbes and Lee Trundle cancelled out Nathan Tyson's 22nd minute opener for the Reds.
But this was a much improved performance from Forest, who on another day could well have snatched all three points themselves.
Having also surrendered in disappointing fashion to Barnsley in their previous outing at the City Ground, there was talk of protests against manager Megson.
But, instead of jeers there were cheers early on as, following a bright start, Tyson fired Forest into the lead.
The former Wycombe man was sent clear by Jack Lester and confidently converted his fifth goal of the season for the Reds, firing a low shot under the advancing Willy Gueret.
By that stage Swansea will have felt they should have been ahead themselves after a mazy run from fit-again Trundle saw him skip past three players in the Forest box before striking a low shot that cannoned back off the foot of the post.
But they were not behind for long as substitute Adrian Forbes - brought on after an early arm injury ended Steven Watt's participation in the game - showed a sublime touch to net an equaliser.
Swansea skipper Roberto Martinez sent a fine pass over the top of the home defence and the winger showed tremendous pace to beat keeper Paul Gerrard to the ball and lift a fine finish over the keeper and into the back of the net.
In first-half injury time a slip from John Thompson almost let in Andy Robinson, but the winger fired wide of an exposed net with Gerrard off his line.
Another quick attack led by Tyson saw Forest pressure again but Kris Commons saw a fierce shot charged down.
Commons himself then tried his luck with powerful 18-yard shot that Gueret did well to save at his near post at the expense of a corner.
The home side came even closer as Lester dived bravely to connect with a Julian Bennett cross at the near post, but saw his effort bounce inches wide with Gueret scrambling across goal to cover.
But despite all the Forest pressure it was Swansea who were to snatch the lead again through a moment of individual brilliance from Trundle.
The striker picked up possession just inside the box on the left side and showed quick feet to engineer the room for a shot which he struck with great accuracy into the top corner of Gerrard's net.