Villa struck first blood early in the first half, as Andreas Weimann slipped Gabriel Agbonlahor in behind the Swansea defence, leaving the Villa striker to dispatch the one-on-one calmly past visiting keeper Gerhard Tremmel.
Despite the setback, Swansea maintained their passing ethos, a move which eventually paid dividends as Roland Lamah headed home at the back post from Pablo Hernandez's cross.
Paul Lambert's men eventually held firm to record a long-awaited point, however, the draw does no favours for either side in a looming relegation battle.
Neither side came into the game in the richest vein of form, with Villa losing their last four in the league and Swansea also without a league win in four.
Though the hosts sat in 13th and the Welsh side two places above them in 11th, both sides entered the game very much looking over their shoulder at the relegation trapdoor, sitting just three and four points above 18th place respectively.
Villa, coming off the back of a Boxing Day defeat against Crystal Palace, shot out of the blocks early as Agbonlahor's pace and strength caused plenty of problems for Swansea's backline.
An early cross from Leandro Bacuna after great work from Fabian Delph gave Swansea an early warning, one that they failed to heed soon after.
On seven minutes, Weimann and Agbonlahor combined excellently to allow the Austrian to play a slide-rule pass into the feet of the Englishman, who made no mistake to score his first goal since Villa's last victory, a 3-2 win against Southampton earlier this month.
In fact, Agbonlahor's goal was Villa's first in the first half at home all season in the Premier League, and gave them the footing from which they could sit back and soak up the pressure from Swansea's passing game.
Swansea sought to take advantage of their domination in possession early on, and they nearly did through defender Ben Davies, but the left-back's header from Jonathan de Guzman's corner bounced off the turf and agonisingly over the bar.
Michael Laudrup's side endeavoured to equalise throughout the remainder of the first half, and de Guzman, Jonjo Shelvey, Hernandez and Wilfried Bony all shared attempts as Villa stood firm.
De Guzman came closest on 34 minutes, lashing a loose ball just wide on the volley from 18 yards out after Antonio Luna's poor defensive header.
Swansea eventually bagged a deserved equaliser two minutes later, although the goal was from an unlikely source.
Shelvey picked out Hernandez wide on the right, whose back post cross was perfectly weighted for the waiting Lamah, who headed home to score his first Premier League goal.
The goal proved the perfect tonic for Swansea, who were dominant in possession before but lacked a true cutting edge.
Bony's swivel and shot just one minute after the goal was followed quickly by an excellent exchange between Shelvey and Hernandez, but the Englishman could not quite retrieve control of the ball as Villa managed to see out the first half with no more damage done.
Villa briefly grabbed a foothold in the game early in the second half, as they looked to stop a losing run of four games in the league.
The visitors quickly reassumed their stranglehold, however, and Brad Guzan had to be alert to keep out a swirling de Guzman free-kick on 52 minutes.
Just over 10 minutes later and another chance fell Swansea's way, as Shelvey cut back inside before curling an effort towards goal. Guzan was helped by a deflection, however, and Villa survived yet another chance.
They nearly grabbed all three points at the death as Ashley Westwood laid off to Delph, who unleashed an effort which stung the palms of Tremmel from 30 yards out, but neither side could grab the crucial winner.