The Eagles, floundering in the drop zone after being docked 10 points for going into administration, put in a spirited display and were deservedly level when Clint Hill cancelled out Gabor Gyepes' early opener.
But Cardiff pocketed another three points towards their promotion bid when Burke smashed a second-half winner.
Palace were missing six first team players through injury and suspension and it took the Bluebirds just four minutes to breach their patched-up defence.
Peter Whittingham's low corner was helped on at the near post by Stephen McPhail and Gyepes was on hand to fire his first goal of the campaign.
Palace are nothing if not resilient, though, and they almost hit back in the 19th minute when Hill's header from Darren Ambrose's free-kick beat the dive of David Marshall but came back off the crossbar.
Marshall did well to palm away a close-range volley from Ambrose five minutes later, but the goalkeeper suffered a case of the jitters from the following corner.
He came too far for Calvin Andrew's cross and palmed the ball to the feet of Ambrose, but the midfielder's lob landed on the roof of the unguarded net.
Cardiff could have doubled their advantage before the interval when Kelvin Etuhu broke clear but the on-loan Manchester City striker's shot was too close to Julian Speroni, who saved with his shoulder.
After the break Ambrose missed a gilt-edged chance to level when he sidefooted Johnny Ertl's cross wide.
And Palace were screaming for a penalty when Tony Capaldi pulled down Stern John at a corner, but referee Carl Boyeson was not interested.
The goal they had been threatening finally arrived just before the hour mark, albeit from an unlikely source. Ambrose swung in a corner and Hill's downward header beat Marshall for his first goal of the season.
But City wrapped up the win 10 minutes later when Palace failed to deal with Mark Kennedy's free-kick and Burke hammered his volley past Speroni from the edge of the area.
Palace had a chance to level in stoppage time but Sean Scannell's shot was deflected wide as the Eagles slumped to a sixth defeat in seven home games.