The Bluebirds had qualified for the tie by virtue of a midweek replay win over Bristol City, and Jones said: "We have been seriously hit with a virus but have had to dig deep for two victories.
"At 2-1 down I couldn't see where we could get any movement to be back in the game but we had the breaks and we took them."
Jay Bothroyd put the Bluebirds ahead in the 17th minute with a splendid diving header from a Michael Chopra cross but the visitors then took control and it was no surprise when Michael Morrison and Dany N'Guessan put their side in front at the interval.
For much of the second period the Foxes looked most likely to add to their score until Peter Whittingham struck home a 72nd-minute free-kick to level the scores with his 17th goal of the season.
Cardiff, who only had six fit players for the substitutes bench had put on Burke in the 61st minute and he began tormenting the visitor's defence down the right flank.
With 90 minutes on the clock he fired in to put the Bluebirds ahead and, with the cheers still ringing around the ground, McCormack slotted home after another Burke effort had been pushed out.
"I think we deserved better than that but you have to take your chances," said Leicester manager Nigel Pearson, who was angered that an apparent trip on Dany N'Guessan did not yield a penalty.
"By half-time we were well on top but you cannot concede goals like that.
"Our set-plays were good, their keeper made some great saves and we were denied a clear penalty but to lose 4-2 is poor from our point of view.
"We cannot feel sorry for ourselves and must get back to winning ways in the league as the result today is not a true reflection of how the game went."
Cardiff manager Jones is still hoping that funds will be made available for him to strengthen the squad.
"When I get the green light I will bring some players in and we will start to fly," he said.
"We are in a results-based business and when we get the right results we will see the benefits as this club is right for investment.
"We will take whatever comes and it is important that we have good results for the football club, the board and those people who are looking to buy in to Cardiff City."