But Hart was angry with referee Carl Boyeson, who failed to give a penalty when Tony Capaldi hauled down Stern John and then harshly ruled handball against Neil Danns, giving Cardiff the free-kick from which Burke hit the winner.
"You have to look at a blatant penalty - how it's been missed I'll never know," said Hart.
"And then a handball when the ball has been battered at the player from a yard. Up to that point there was only one team in it."
Palace were missing six first-teamers through injury and suspension and it took the Bluebirds just four minutes to breach their patched-up rearguard.
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, and they eventually converted one of a host of chances they created when Hill's downward header from Darren Ambrose's corner beat City goalkeeper David Marshall.
But they slumped to a sixth home defeat in their last seven matches when ex-Eagle Mark Kennedy swung in the contentious free-kick and the ball fell to Burke to thump home from the edge of the area.
"I think it was a magnificent performance," added Hart.
"We are patching players up and we have kids on the bench - we are what you see.
"We've got seven games to go and if we play like that we will win football matches. The players are disappointed and angry."
Palace's administrators are in talks with a potential buyer for the cash-strapped club - one piece of good news on another grim day for the club.
"I'm led to believe we are getting closer, which is brilliant," admitted Hart.
Cardiff boss Dave Jones was relieved to pocket another three points towards their play-off bid.
"The ball was in the air a long time, it was difficult to get the ball down and play, and I thought we dealt with it as best we could," he said.
"At this stage of the season, sometimes you have got to win ugly."