The Iceland midfielder's stunning strike proved to be the difference as Lukasz Fabianski kept the visitors at bay with some fine saves
The Welsh club had not won their first post-Christmas fixture since a 4-1 triumph over Cheltenham Town in League One seven years ago, but welcomed a Villa side who had emerged triumphant in just three Boxing Day fixtures in the Premier League era.
Swansea were dominant in the first half at the Liberty Stadium and opened the scoring with a well-struck Gylfi Sigurdsson free-kick after 13 minutes.
The home side continued to look the more likely to score after going ahead, although the introduction of Andreas Weimann at the break gave Villa more of an attacking edge.
Despite an improved second-half display from Villa, which included injury-time chances for Christian Benteke and Gabriel Agbonlahor, Garry Monk's men were able to hold firm for the three points and a first Boxing Day win in the Premier League.
Top scorer Wilfried Bony returned to the Swansea starting XI as one of four changes from the 1-0 win at Hull City last weekend, while Paul Lambert brought in Alan Hutton and Tom Cleverley following suspension.
Matches between these sides have become a staple of the festive calendar since Swansea's promotion to the Premier League in 2011, but the hosts did not appear to be in a charitable mood early on as Jefferson Montero headed over from Nathan Dyer’s cross in the opening minutes.
Agbonlahor, who had his red card for a collision with Manchester United's Ashley Young overturned this week, sent the ball whistling past the right-hand post from 20 yards at the other end - although Swansea went ahead through Sigurdsson after 13 minutes.
The Iceland international - also back in the team after missing out through injury at Hull – wrong-footed goalkeeper Brad Guzan from a 20-yard free-kick, awarded when Jores Okore felled the advancing Montero.
Swansea continued to apply the pressure after breaking the deadlock, yet the dangerous Montero was forced off with an apparent hamstring injury after 20 minutes.Aside from Agbonlahor's early effort, Villa's first-half chances were restricted to a Benteke free-kick that crept just wide having found a gap in the Swansea wall shortly after the half-hour mark.
Ashley Williams will have been grateful to see the offside flag when he headed wide from close range before the interval, and Villa boss Lambert responded to the pressure by bringing Weimann on for Aly Cissokho.
The visitors looked a different team after the Austrian's introduction, although his first real strike was comfortably collected by Lukasz Fabianski and his cross soon after was played too close to the Polish goalkeeper with three Villa shirts lurking dangerously in the penalty area.
Bony had the ball in the net after latching onto a throughball, only to see the effort rightly ruled out for offside, before Agbonlahor chipped wide under pressure from Williams with appeals for a penalty turned down by referee Roger East.
Dyer sent a limp effort into the arms of a grateful Guzan late on, while Swansea also saw their own appeals for a spot-kick waved away after the ball appeared to strike the hand of Ron Vlaar, but two superb stoppage-time saves by Fabianski from a Benteke header and Agbonlahor poke ensured maximum points for the hosts.