The Belgian's first ever Premier League goal was enough to earn a point for Michael Laudrup's men and continue Paul Lambert's woeful record at Villa Park.
Gabriel Agbonlahor broke the deadlock amid some questionable Swansea defending in the seventh minute at Villa Park.
But the visitors totally dominated possession for the remainder of the half, with Roland Lamah heading home a deserved 36th-minute leveler.
Despite maintaining that pressure in the second period, Swansea was unable to net a winner as the spoils were shared in a fitting bookend to 2013, with the sides having drawn 2-2 at the Liberty Stadium on Jan. 1.
Villa and Swansea have now both gone five top-flight games without a victory.
Paul Lambert made three changes to the Villa side that lost 1-0 to Crystal Palace on Boxing Day, with Ashley Westwood, Leandro Bacuna and Aleksandar Tonev coming in for Karim El Ahmadi, Marc Albrighton and Libor Kozak.
Swansea, meanwhile, made seven alterations to the XI that suffered a 1-0 defeat at Chelsea on Thursday, with Lamah, Wilfried Bony and Pablo Hernandez among those included in the starting lineup.
Following an open start to the game, Agbonlahor grabbed his second Premier League goal of the season as Villa easily carved through the Swansea defense.
Andreas Weimann knocked the ball beyond Chico Flores to send Agbonlahor clean through on goal, and the striker made no mistake in tucking away a neat finish past Gerhard Tremmel.
Hernandez had the visitors’ first clear-cut goalscoring opportunity 11 minutes later, poking the ball between the legs of Matthew Lowton before seeing a low drive from the edge of the penalty area stopped by Brad Guzan.
Michael Laudrup's men dominated possession as the half progressed, with Jonathan de Guzman sending a left-footed half-volley just wide of the post 10 minutes before the interval.
But Swansea did equalize just a minute later when Lamah headed home at the back post from Davies' superb cross from the left - the on-loan Belgian's first Premier League goal.
De Guzman forced Guzan to tip over the crossbar from a 25-yard free-kick early in the second half as Michael Laudrup's team maintained its pressure from the opening 45 minutes.
However, for all of Swansea's possession, Guzan remained relatively untroubled for large parts of the half as the visitors struggled to deliver in the final third.
Agbonlahor fired a 70th-minute effort high and wide as Villa went in search on an unlikely winner, but chances remained few and far between for the hosts.
Laudrup threw on Alvaro Vazquez in place of Bony in the closing stages, but the Spaniard failed to make an impact as both teams ended the year with a draw.