Clarets manager Brian Laws named an unchanged team from the side that had lost to Wigan Athletic while Dunn shrugged off the injury he picked up in the midweek victory over Birmingham City to start.
The only goal of the game arrived from the penalty spot after referee Mike Dean had adjudged that Martin Olsson had been brought down by the legs of Burnley goalkeeper Brian Jensen.
Jensen was not even shown a yellow card for the offence but Dunn stepped up and confidently placed his penalty high to the goalkeeper's left.
At the other end of the pitch, Burnley's furious penalty claims were waved away after Wade Elliott was brought down right on the edge of the box by Christopher Samba.
The nearest Burnley came to an equaliser before the interval came from a quickly-taken Graham Alexander free-kick from which Leon Cort volleyed wide.
Burnley suffered a blow when left-back Stephen Jordan limped off to be replaced by Danny Fox.
Olsson had two striking opportunities in the second half, the first of which he was unfortunate to see a shot cannon off the underside of the crossbar. His next strike brought a fine save from Jensen.
Blackburn goalkeeper Jason Brown survived strong claims that he had carried the ball outside his penalty area but, despite plenty of possession, the home side struggled to create goalscoring opportunities in front of goal with the Rovers defence determined to protect their lead.
The nearest Burnley came to an equaliser came when substitute Fox made a fine overlapping run down the left but his teasing cross evaded strikers and defenders alike.
The result means that Sam Allardyce's team have completed the double over their East Lancashire rivals but for the Clarets it's now a case of one foot already in the Championship as their Premier League dream increasingly looks like lasting for only one season.