Referee Mike Dean then failed to spot an obvious handball by Rovers keeper Jason Brown as Burnley's 31-year wait for a win over their neighbours continued.
Burnley's fans showed plenty of passion, but Brian Laws' players failed to rise to the occasion and remain three points adrift in the Premier League drop zone.
Sam Allardyce's Rovers, meanwhile, move up to 10th in the table with their second victory in five days.
Burnley were awarded a free-kick early from which Graham Alexander sent in a curling delivery that Leon Cort was just unable to get a head to.
Steven Nzonzi then rose in the box at the other end but failed to connect with a Morten Gamst Pedersen free-kick before the Norwegian midfielder fired a wild effort high and wide from long range.
Blackburn thought they had the lead in the 18th minute but Olsson was ruled offside after taking the ball down, turning and lashing it past Jensen.
If that decision appeared contentious, even more controversy surrounded the incident that followed as Rovers made their breakthrough.
Olsson broke into the box and cut inside Jensen, who had rushed out to meet him.
There seemed to be minimal contact between the two players but Olsson tumbled to the turf and Dean pointed to the spot.
Dunn stepped up to smash home his third goal of the week and put the visitors ahead.
Burnley tried to raise their game and Wade Elliott put a cross into the danger-zone which Chris Samba headed behind for a corner.
Blackburn looked to hit back and Roberts twisted and turned his way into the box before being crowded out.
Dean had done little to enamour himself to the Burnley faithful and made himself even more unpopular in the 39th minute as Samba brought down Elliott on the edge of the area.
Samba was booked and Clarets fans cried out for a penalty but Dean only gave a free-kick which was subsequently wasted.
Both sides had chances to score just before the break with Michel Salgado blasting wide and then Cort failing to guide his effort on target having brought the ball down on his chest.
Burnley started the second period brightly and Steven Fletcher was almost quick enough to take advantage when Nzonzi under-hit a back-pass, but Brown came out to clear.
Tyrone Mears then brought down a ball flighted into the box but Olsson put it behind for a corner before Mears could get a shot away.
The Clarets finally registered a shot on target in the 54th minute as Martin Paterson turned and fired from distance into the arms of Brown.
At the other end, Cort and Jensen almost got in each other's way in scrambling to clear a dangerous Pedersen free-kick, but Burnley's supporters had been lifted and roared their team on to keep attacking.
Paterson made way for Robbie Blake and Alexander attempted a speculative volley which went well wide before being booked for showing his studs to Pedersen.
The resulting free-kick was not properly cleared and Olsson rifled an effort that cracked against the crossbar and bounced down, possibly over the line, before Jones headed the rebound wide.
Olsson then found himself unmarked in the box, but Jensen parried his effort after he had controlled and shot.
Salgado whipped in a volleyed cross which substitute Franco Di Santo headed off-target.
Dean was involved in yet more controversy with 15 minutes from the end when he failed to spot a clear handball by Brown.
David Nugent chased a ball which Brown rushed out to gather, but the goalkeeper made a hash of it and clearly handled as he stumbled out of the box.
The crowd were incensed but Dean missed it and play went on.
Burnley were inches away from an equaliser soon after when Danny Fox's low cross was narrowly missed by both Elliott and Fletcher sliding in.
Five minutes of added time were played but a disappointing finale completed a miserable afternoon for the home fans as Rovers hung on for the victory.