The Clarets appeared on course to end their 34-year wait for a win over their local rivals when Jason Shackell converted Ross Wallace's 32nd-minute free-kick at the second attempt.
Rovers pressed for an equaliser but struggled to find a way through the Burnley rearguard even after Ben Mee saw red in the 64th minute following two quickfire bookings.
However Dunn, making his first Championship start since October due to persistent calf problems, netted in the fifth of six additional minutes when he drove a shot past Lee Grant after Cameron Stewart's chip was touched on by Jason Lowe.
The match was interrupted early on by a chicken which had been released onto the pitch by a Blackburn fan in protest against the club's poultry magnate owners Venky's.
The hosts pushed forward after the unscheduled stoppage and striker Jordan Rhodes was left with a bloodied nose when he took a boot to the face from Burnley centre-back Michael Duff.
Rovers goalkeeper Jake Kean was then called into action to keep out Martin Paterson and Wallace before Shackell broke the deadlock.
The visitors instantly looked to press home their advantage with Rovers reeling, and Wallace came agonisingly close with 30-yard free-kick that clipped the upright.
The home side struggled to create many chances in a poor game, and even Mee's dismissal did not act as an impetus to push on for Michael Appleton's men.
Just as Burnley were starting to look comfortable, Mee's heavy touch on a run forward from left-back saw him career into Grant Hanley and earn a second booking six minutes after a late challenge on Dunn.
The Clarets' defence was still largely untroubled and Blackburn looked down and out until the final few moments when midfielder Dunn pounced to rescue a point.
The draw means that Burnley stay in 13th place, level on points with Derby a spot above. Blackburn however move up one place above Ipswich, pulling themselves a little further away from the Championship drop-zone.