A free-kick by James McFadden briefly put Blues level but they fell to another defeat on a ground they last won at back in 1968.
Rovers, looking to equal their best run of the season of seven home games unbeaten, were unchanged from the heroic 1-1 draw with Chelsea on Sunday.
Skipper Ryan Nelsen returned to the bench after three games out with a knee injury.
Blues boss Alex McLeish brought Lee Bowyer back into midfield and James McFadden into attack following their 3-1 defeat at Sunderland on Saturday.
Blackburn got off to a dream start as Dunn fired them into a fifth minute lead.
A long ball from Chris Samba was touched down by Nikola Kalinic and Dunn expertly placed a low left-foot drive past Joe Hart in the Blues goal from 18 yards.
Soon afterwards, a cross from the left by El Hadji Diouf was just too high for Kalinic in the goalmouth but, at the other end, Rovers keeper Jason Brown had to be at his best to keep out a shot from McFadden which deflected off Diouf.
Brown was in fine form again to get to a cross from Stephen Carr just ahead of Cameron Jerome as Blues continued to put on a spirited show.
But three minutes before the break, Dunn had the ball in the net for Blackburn again only for it to be ruled out due to a foul by Michel Salgado on Bowyer.
Rovers replaced Kalinic with Jason Roberts at the start of the second half in a bid to rediscover the formula which brought Sunday's equaliser against Chelsea.
But 11 minutes after the restart, Blues were level as McFadden steered a left-foot free-kick from 20 yards well past the right hand of the stranded Brown.
The kick was awarded for a foul on Roger Johnson by Morten Gamst Pedersen.
Rovers came back strongly and retook the lead on 68 minutes when Dunn scored his second of the game.
As Diouf's corner came over from the right, he beat Hart with a header at the near post.