The pressure on Pedro Caixinha grew on Wednesday night as the Gers failed to get back to winning ways following their Scottish League Cup semi-final defeat to Motherwell at the weekend.
And it appeared that they would get themselves back on track - just - as they went into stoppage-time at the end of the second-half 1-0 up thanks to a first-half goal from Jason Holt.
However, the game ended in frantic fashion. Rangers were awarded a penalty but Ryan Jack was sent off for the home side before Daniel Candeias could take the spot-kick, which he subsequently missed, before Killie went down the other end and Burke - who spent just short of a decade on Rangers' books earlier in his career - scored the equaliser for the visitors.
It was a poor result for the Light Blues, who have now fallen to fourth in the Scottish Premiership eight points off the summit after just 10 games. But Burke has suggested that had Steve Clarke's men - who were bottom going into the contest - got what they deserved, it could have been even worse for Caixinha.
“Everybody’s a little bit on edge and on their seat because they’re not killing the game off. But I think we gave as good as we got. We were terrific, especially the second-half. We showed a lot of belief, took a lot of positives from the game and I think in the end we have to be a little bit disappointed we didn’t take the full three points," he said, as reported by The Scottish Herald.