A last-gasp equalizer gave the Welsh side a deserved point at the Liberty Stadium as the visitors continued a worrying run without a victory.
Goals from Andreas Weimann and a late penalty from Christian Benteke appeared to have sealed a comeback victory for Villa, who had fallen behind to an early goal from Wayne Routledge.
But a last-gasp strike from Graham saved a deserved point for Swasnea, which had dominated much of the match and missed a number of goalscoring opportunities.
With the busy festive schedule hitting both sides, managers Michael Laudrup and Lambert both rung the changes.
Swansea made five alterations, with fit-again Michu among the players returning to the lineup while Villa made four, naming its youngest ever starting XI for a Premier League match.
It was almost 1-0 within three minutes when Routledge got the better of Chris Herd and was clean through on goal before he failed to beat Brad Guzan one-on-one, the keeper staying big to make the block.
Routledge made no mistake moments later though, when he rounded Guzan after receiving Pablo Hernanez’s pass and fired home from a tight angle
Michu struck the inside of the post and Danny Graham somehow failed to hit the target from eight yards with a presentable chance as the Swans absolutely dominated the early stages.
The club should have scored on numerous occasions and Michu again hit the woodwork after a sharp turn and shot saw him find a post for the second time after good work from Routledge and Ashley Williams.
As the half progressed, Villa slowly gained a foothold in the game, but the equalizer on the stroke of halftime still came as a surprise.
A good ball from Benteke and Williams’ missed interception gave Weimann the chance to net and he took the chance brilliantly with a superb first-time strike to get the better of Michel Vorm.
After the break there were less clear-cut opportunities, although Guzan had to save strongly from a Michu header on the hour mark.
At the other end, an acrobatic attempt by Benteke deflected into the path of Weimann, who almost found the target to complete an unlikely turnaround in the match.
That breakthrough did come with just six minutes remaining. Benteke’s header was cleared off the line by Nathan Dyer but as the rebound Weimann, the Swansea man fouled and conceded a penalty.
Benteke stepped up to send Vorm the wrong way from the spot and it looked like the visitors had held on to secure three points.
However, in the fourth minute of stoppage time Dyer crossed into a chaotic penalty box and Graham stayed calm to slot home a deserved equalizer.