Jay Bothroyd had put the home side in front but Michael Morrison and Dany N'Guessan turned the game on its head.
Peter Whittingham's 17th goal of the season drew Cardiff level 19 minutes from time and with time running out Burke and then McCormack's first goal of the season stunned Leicester.
The Foxes had an opportunity to take the lead after only two minutes but striker Martyn Waghorn fired over from the edge of the penalty area.
And Cardiff breathed a sigh of relief once again five minutes later when N'Guessan flashed a shot wide from 12 yards out.
But it was Cardiff who opened the scoring after 17 minutes when their strike duo of Michael Chopra and Bothroyd combined to great effect.
Chopra was the architect, crossing from the right for the onrushing Bothroyd who made no mistake with a downward header from six yards out.
But Nigel Pearson's side showed a positive response to going behind and they were level after 33 minutes when Paul Gallagher's free-kick picked out Morrison and the defender slotted home with a neat right-footed finish.
And the visitors were in front five minutes later when Waghorn found N'Guessan and the former Lincoln man made no mistake with his close-range finish to stun the Bluebirds.
The lively N'Guessan was brimming with confidence and he tried his luck from just outside the penalty area but his curled effort was wide of the target on the stroke of half-time.
Midfielder Matt Oakley was presented with a sight of goal five minutes after the break but the former Derby man dragged his shot wide from the edge of the area.
Chopra forced Leicester keeper Chris Weale to palm away his right-footed strike as Cardiff chased and equaliser.
Yann Kermorgant almost gave Leicester some breathing space but his left-footed effort flashed wide.
And the Foxes were left to rue their missed opportunities when Whittingham drew Cardiff level with 19 minutes remaining.
The prolific midfielder swung in a free-kick from the right which evaded everybody and found the far corner of the net.
And the home side had an opportunity to retake the lead moments later but winger Burke fired over the bar from 20 yards out.
Both sides were clearly eager to avoid a replay and it took a fine save from David Marshall to keep out Kermorgant's close-range header.
But with time running out Burke made no mistake with a right-footed finish from 12 yards out and McCormack put the icing on the cake when he slotted home from close range after being set up by Bothroyd.