Charles N'Zogbia's early strike looked set to leave Villa a healthy margin between themselves and the drop zone, but Dunn's 85th-minute header means there is still a chance of them being dragged into the relegation fight.
Villa now head into a pair of crucial home games against Fulham and Bolton in the knowledge that points are imperative with the likes of Chelsea and Liverpool awaiting next month.
McLeish said: "I don't think we can afford to be complacent. Last year we were around the same position, maybe lower. We've got to believe we can climb the table and that's the challenge for the whole team.
"All I'm looking to is next week. We've got to concentrate on the next game and not about things that are too far in front. You've got to think about the particular challenges that are coming up."
After dominating the first half, in which Blackburn goalkeeper Paul Robinson made a series of stunning saves to keep his side in the game, Villa were pegged back after the break and were ultimately fortunate to escape with a point.
Marc Albrighton missed an early chance for Villa before N'Zogbia opened the scoring with a fine left-footed drive into the roof of the net after being sent through by an excellent threaded pass from Carlos Cuellar.
N'Zogbia saw his chance to increase Villa's lead denied by Robinson on the stroke of half-time, and in a portent of what was to come Rovers ended the half with a chance of their own when Junior Hoilett headed over from six yards.
Scott Dann headed on to the roof of the net in the 55th minute, Junior Hoilett had an effort parried by Shay Given, then the Villa keeper reacted superbly to parry a drive from Morten Gamst Pedersen.
Robinson was called into action to deny Gabriel Agbonlahor before Rovers grabbed their deserved equaliser through substitute Dunn, who headed past Given from point-blank range.
Dunn's half-time introduction had fired up Rovers after what boss Steve Kean admitted was a dismal first-half display - and Kean acknowledged the value of his home-grown star in their ongoing relegation fight.
Kean said: "David was disappointed he could not play the whole game because he picked up a nasty little thigh strain, but the medical team told me he was good for the second half.
"David is Blackburn through and through, and his whole family have been Blackburn fans all their lives. He knows what it means to us to score goals, and it was a very important one for us."
Like Villa, Rovers now head into a series of games which look winnable on paper, and Kean added: "It was an important point for us because we have made a bit of ground on the other teams. It wasn't the win we would have liked, but it keeps us going."