The Royals went 1-0 up against the run of play in the 31st minute when goalkeeper Michel Vorm allowed Pavel Pogrebnyak's shot to slip beneath him and in.
They then doubled their advantage a minute before the break as Noel Hunt beat Vorm with a tidy volley, but Swansea continued to dominate play in the second half and after Michu converted a 71st-minute header, Wayne Routledge's fierce strike drew the contest level in the 78th minute.
Overall, Reading were lucky to come away with a point in the end, with the hosts being thwarted by a succession of excellent saves from Royals stopper Alex McCarthy - who twice sent efforts onto the post - and also having a strong-looking penalty claim for handball dismissed towards the end.
It did take a goal-line clearance at the death from Swansea captain Ashley Williams to keep out Jem Karacan's header, though, and Reading will feel a fair degree of disappointment at missing this opportunity to end a winless top-flight streak which now extends to six games.
The Swans, meanwhile, have stopped their rot, having lost their previous three Premier League games going into the match.
The Welsh outfit had made a bright start, creating several decent chances in the opening 20 minutes.
Williams - named this week as the new Wales skipper - nodded off target three times while Danny Graham saw a header turned over the bar by McCarthy, who also batted away a shot from Routledge.
Routledge then played a one-two with Nathan Dyer and struck an effort that was blocked by Adrian Mariappa, with replays suggesting a suspicion of handball.
Reading had shown little going forward, but just after the half-hour mark they snatched the lead as Pogrebnyak latched on to a through-ball from Jobi McAnuff and hit an effort at Vorm.
It should have been a routine save for the Dutchman but instead, it somehow sneaked beyond him and into the net.
Swansea looked to make a swift response and Ki Sung-yueng drilled into the arms of McCarthy from distance, but in the 44th minute Reading struck again.
A McAnuff shot looped up off Angel Rangel and Hunt was on hand to lash the ball past Vorm.
Swans boss Michael Laudrup reacted by substituting Graham at the interval for Luke Moore, who was straight into the thick of the action.
First, he found himself through one-on-one with McCarthy but was denied by the goalkeeper's foot.
Moore then sent a header goalwards which McCarthy did superbly to push against the post.
McCarthy repeated the trick soon after by pushing Ki's deflected low shot onto his other upright, and then got down well to save a Williams header.
He was finally beaten with 19 minutes to go as Michu powered in a diving header from a deflected Routledge cross.
McCarthy pulled off yet another great stop to keep Moore out, but Swansea were not to be denied and after Ki robbed Reading of possession, a forward break was rounded off by Routledge's firm finish.
The hosts cried out for a penalty when Nicky Shorey appeared to block Pablo Hernandez's shot with his arm in the box, but play went on and the late drama continued as Williams jumped on the line to nod away Karacan's header after another uncharacteristic Vorm error.