A quiet first half in South Wales saw Villa actually impress in parts of it, Rudy Gestede providing a solid outlet up front. However, Guzan's error allowed Federico Fernandez to head into an empty net, clinching all three points for Swansea.
The win leaves Villa 12 points from safety with just seven games left, having picked up just 16 from 31. Swansea are up to 15th, and now have some breathing space.
Villa make something of a bright start
Villa manager Remi Garde resisted the temptation to blood some youngsters, picking an experienced line up despite the wishes of many fans to see some young talent.
They couldn't really argue with the team selection based on Villa's start, as they were the better team, frustrating the crowd by pegging Swansea back. Rudy Gestede struck wide just four minutes in, before Jordan Ayew rifled over the bar after pick-pocketing Leroy Fer in midfield.
Once Gestede was done tussling with Ashley Williams, the pair scrapping after falling to the floor challenging for a long ball. He'd be denied by Williams later in the half, the centre-back heading off the line from Gestede's overhead kick.
Swansea grew into the game as the half progressed, Bafetimbi Gomis hammering wide after rolling Ciaran Clark on the edge of the area. Clark himself went close on a couple of occasions, heading over from corners.
Fernandez puts Swans ahead
The home side did get themselves into the lead eventually, punishing a costly mistake by Guzan just eight minutes into the second half. The 'keeper came to claim Gylfi Sigurdsson's free-kick, only to miss the ball and watch on in horror as it bounced off Fernandez and into the net.
Poor response from Villans in final period
Villa fans have been subjected to conceding the first goal in most of their games this season, but it's the response to going behind that would have deeply frustrated them in Wales, with Garde's men showing no sign of equalising.
It took them until over 20 minutes after Swansea's goal to get their second, and final, shot on target of the game, Gestede having an effort blocked after being set up by young substitute Andre Green.
Swansea looked more like grabbing a second than Villa did equalising, and Gomis came close to netting in the dying minutes. He'd held the ball up well all game, and did so once more, nutmegging Aly Cissokho before blasting just wide.
Villa must now move on as their long goodbye to the Premier League enters it's final few weeks, they host Chelsea next Saturday.