The former Scotland winger found the net on the cusp of half-time to hand Angelo Alessio's charges just their second Ladbrokes Premiership victory of the campaign and send them into the top six ahead of the international break.
Aside from a 3-1 defeat against Celtic a fortnight ago, Kilmarnock's on-loan Juventus goalkeeper Laurentiu Branescu has not conceded from open play since George Oakley scored in a 2-0 win for Hamilton on August 10 – a run of six shutouts from seven outings.
And injury-ravaged Hearts, who were without Uche Ikpeazu due to a hamstring complaint and initially lined up without a recognised striker, were unable to breach the visitors' rearguard.
The Gorgie outfit now lie just two points off the bottom and the full-time whistle was met by a cacophony of jeers from the disgruntled home supporters.
The recent history of this fixture would have given Killie cause for optimism. Six of the previous eight meetings between the teams ended in away wins, including all four encounters last season.
And it was Alessio's men who were first to go on the offensive, with Liam Millar heading a superb Burke delivery narrowly over the bar.
Loic Damour threatened for Hearts with a fizzing low drive from the edge of the box but it was saved by the in-form Branescu.
The Jambos had a glorious opportunity to take the lead moments later when Glenn Whelan met a low cross by Clare, only to fire wastefully over the bar from 14 yards.
Levein's men paid for that profligacy three minutes before half-time when Burke ghosted in from the right flank to head home a wonderful delivery from NiKo Hamalainen, via a deflection off Jake Mulraney.
Eamonn Brophy was presented with a passable chance to put the game to bed in the second period when Millar scampered down the left before finding the striker in the box, however he could not get his shot away under pressure from Christophe Berra.
Hearts' on-loan Manchester City forward Ryotaro Meshino, a consistently lively presence in the final third, stung the palms of Branescu after jinking into space on the edge of the box.
Meshino, Ollie Bozanic and Clevid Dikamona all saw half-chances miss the target in the dying embers, while Steven MacLean did ripple the net but referee Alan Muir had already pulled the play back for a foul.