Danny Graham levelled matters four minutes into added time, as the Swans salvaged a 2-2 draw.
Wayne Routledge fired the hosts ahead in the ninth minute, with Andreas Weimann netting Villa's first just 60 seconds before the break.
Christian Benteke's penalty on 84 appeared to give the struggling visitors a welcome three points only for Graham to secure a late point for the Welsh side.
Laudrup admitted: "We had a fantastic first half but there was only one problem - it was 1-1 at the break when we should have been 4-0 ahead inside 10 minutes.
"We created the chances, played so well in the first 40 minutes but did not score. We did not get that second goal which could have led to the third and the fourth.
"But we came back and showed character and scored in the last minute. The title you can give the game is 'we should have won it and could have lost it'."