Liam Scales scored the only goal at Celtic Park in a frustrating day in front of goal for Martin O'Neill's men.
Record signing Kasper Hogh's goal drought stretched to 16 games after he had an early opener controversially ruled out for a foul by Mika Baur in the build-up.
Scales broke the deadlock when the Irish defender's finish from Camilo Duran's cross looped into the corner.
Baur also had a goal disallowed for offside in the second period before Benjamin Nygren saw a penalty saved by Kilmarnock goalkeeper Nico Mantl.
However, one goal was enough to open up a four-point cushion on Rangers, who have a game in hand.
"I thought we were getting a little bit tired and the penalty would see us through. As it turns out, we miss and that gives Kilmarnock a lift," said O'Neill.
"Overall, just delighted to have won."
There will be no repeat of Hearts' heroic title challenge last season, up against the far greater resources of both Glasgow giants.
The Jambos fell 5-0 down inside 47 minutes against a rampant Falkirk to suffer a first league defeat at Tynecastle since April 2025.
"This is an embarrassment, the first half for sure," said Hearts boss Wouter Vrancken.
"It's crazy, crazy, and everybody - me, staff, players - have to take responsibility in those things. This is ridiculous."
Finley Barbrook, Owen Eames, Liam Henderson and Jacob Murrell's double secured another famous win for Falkirk, who beat Celtic last weekend for the first time in 19 years.
Falkirk move above Hearts into third.