Michael Laudrup’s men were vastly superior in the first half and the home side broke the deadlock courtesy of Wayne Routledge’s ninth-minute strike.
However, it was Villa’s in-form duo Andreas Weimann and Christian Benteke who gave the visitors a shock lead – the midfielder placed his finish under Michel Vorm before the Belgian scored from the spot.
Villa looked to be heading towards maximum points with minutes remaining but Danny Graham’s late volley salvaged a draw for the Swans, leaving Lambert’s men two points above the relegation zone.
“With 30 seconds to go you expect to see it through. First half Swansea were right on top but second half we were very good,” Lambert told the BBC Sport.
“The goal before half time did change it. We have a lot of injuries so you have to give credit to the players. It is a big point for us.
“I’ve got a lot of faith in what we’re doing and we won’t deviate from it. Even after the heavy defeats we were keeping spirits up.
“It will be a bit happier on the trip home considering what has happened. We took a lot of knocks and kickings today but deserved something from the game.”
Meanwhile, Swansea manager Laudrup admitted his frustration to only secure a point despite their dominant first-half performance at the Liberty Stadium.
“We could have been 4-0 up after 12 minutes. Incredible,” added Laudrup.
“The only negative part was conceding a goal in the end of the first half and not scoring a second one. Aston Villa really fought after three losses.
“They wanted to do everything to get a point and got close to getting three points. We could have lost but should have won.”
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