Daizen Maeda and Giorgos Giakoumakis were the goalscorers, and while Ange Postecoglou's side was the favourite coming into the match, it had to withstand pressure from Kilmarnock right from the kick-off, Danny Armstrong causing problems on his right wing.
But the perils of not taking your chances against the Scottish champion became clear on 18 minutes when Australian Aaorn Mooy delivered a wicked free-kick into the box, and Maeda was on hand to deflect it in as the Killie defence was found wanting.
Celtic held that slender lead at the break, and indeed thought it had settled matters four minutes after the restart when Reo Hatate thought he'd scored, only for the video assistant referee to chalk off the goal (VAR).
Maeda had a potential second erased also after another offside call, but VAR's generosity for Kilmarnock ended there when perhaps it could have continued when Giakoumakis clumsily brought down Joe Wright in the box, but no penalty was given.
Barely minutes later, the Greece international was on hand to tap-in Celtic's second and put the result beyond doubt, wheeling away to celebrate as the Hoops fans cheered yet another chance to win some silverware.
