Chris Wood deservedly gave the hosts a 31st-minute lead but Cardiff looked to have been presented with a great chance to equalise moments after the restart when Ben Mee was penalised for handball.
It was assistant Darren Cann who raised his flag but, after a discussion with Dean, the referee changed his mind, leaving Cardiff’s players and boss Neil Warnock furious.
Cardiff had two more strong penalty appeals turned down late in the second half and Wood made sure of the points with his 13th goal of the season in injury time.
Warnock is already facing a Football Association charge for three breaches of the rules after criticising the officials for failing to disallow a Chelsea goal for offside last month and perhaps wisely he chose not to confront Dean on the pitch at the end of the game.
The result leaves Burnley 11 points above their opponents and surely guaranteed another season of Premier League football, while Cardiff are five points behind Brighton, who they play on Tuesday.
Burnley should have taken the lead in only the second minute but Wood headed against a post from close range after Mee had directed Dwight McNeil’s free-kick back across goal.
There was little free-flowing football being played, with Cardiff creating a few potential openings but not troubling Tom Heaton. Josh Murphy’s 20-yard effort that was comfortably left by the keeper was the closest they came.
It took 27 minutes for Burnley to part the Cardiff defence, a clever pass from Wood setting up Ashley Westwood, who shot tamely at Neil Etheridge from 15 yards out.
The pressure was growing, though, and four minutes later the hosts made the breakthrough. Westwood’s free-kick from out on the left was tipped over by Etheridge but from the corner no one picked up Wood and he made amends for his earlier miss by heading in from five yards.
Burnley should have had a second four minutes later as Westwood again found himself clean through after a fine pass from Ashley Barnes but this time his attempted chip was weak and Etheridge saved.
Cardiff players optimistically urged Dean to show James Tarkowski a red card for bringing down Murphy, with Joe Bennett sending the resulting free-kick wastefully over the bar.
Dean had his cards out again on the stroke of half-time after Bennett brought down Barnes and then trod on the Burnley striker’s leg. Barnes squared up to the defender before bizarrely kissing him twice on the nose. Both men were booked.
Cardiff looked a different team after the break and they reacted positively to the penalty furore.
Heaton was relieved to see a fizzing shot from Harry Arter fly narrowly over the bar before Junior Hoilett spurned the visitors’ best chance with a wild effort from inside the box.
Burnley’s first-half dominance seemed a distant memory but they largely kept Cardiff at arm’s length and Mee came close to doubling the lead with a header than was well saved by Etheridge.
The visitors again appealed for a penalty with the minutes ticking down when Gunnarsson tangled with Charlie Taylor. It was certainly clumsy from the Burnley man, who appeared to catch Gunnarsson’s ankle as he tumbled over, but Dean was unmoved.
Yet another penalty appeal came in injury time when Tarkowski appeared to handle a shot from Bruno Ecuele Manga but Dean once again decided not to give a spot-kick.
Then, to rub salt in the wounds, Burnley broke and McNeil’s cross came off Ecuele Manga and spooned up perfectly for Wood to nod in on the goal-line.