Goals from Stuart Findlay, Chris Burke and Eamonn Brophy secured a fourth successive victory in Gorgie for the Ayrshire outfit, while their overall unbeaten run in Edinburgh's west end stretches back to December 2016.
But they held firm to claim a second win on the bounce under Alex Dyer and ensure Hearts returned to the bottom of the Ladbrokes Premiership.
Hearts were on the attack from the outset, with Euan Henderson stinging the palms of Branescu and a Lewis Moore delivery clipping the crossbar following a deflection off Nico Hamalainen.
Branescu was excelling in the opening exchanges and made a fine low stop to deny Halkett after the centre-half rose highest to meet a deep Moore delivery.
However, it was the visitors who claimed the lead midway through the first half. A flat Rory McKenzie corner was inadvertently flicked into the danger zone by Naismith and Findlay rose highest to power a header beyond Joel Pereira.
Things went from bad to worse as half-time approached when Alan Power took advantage of Hearts' risky high line to send Burke scampering through on goal. The winger's finish was sublime, belying a tight angle to chip the onrushing Pereira and double Killie's lead.
Hearts were afforded a way back into the game when Conor Washington raced onto a flick-on by fellow substitute Uche Ikpeazu, only to be flattened by Branescu. Referee David Munro pointed to the spot and Clare slotted home the resulting penalty.
In a moment of madness, Branescu then hurled the ball away as Clare sought to collect it from the net, which was deemed time-wasting by Munro, and the Romanian received his second booking in 60 seconds.
Substitute goalkeeper Jan Koprivec replaced McKenzie and Hearts had a lifeline.
A miraculous comeback seemed on when Halkett found space in the box to meet a Jamie Walker delivery and rattle a superb low finish beyond Koprivec for his seventh goal of the campaign.
But resolute Killie held firm to claim all three points from a thrilling affair.