Killie took the lead with a stunning 40-yard effort from Souleymane Coulibaly in the 32nd minute but the home side responded in kind.
Moussa Dembele scored twice in three minutes to put Celtic in the driving seat.
The score remained 2-1 at half-time and Celtic sprang a surprise in the second half when they replaced goalkeeper Dorus de Vries with Craig Gordon.
James Forrest then extended the home side’s lead. That was the cue for manager Brendan Rodgers to replace Dembele, with one eye on Wednesday’s Champions League match against Manchester City.
Leigh Griffiths, back from injury, replaced the former Fulham striker and had an instant impact, scoring within two minutes to make it 4-1.
He then won a penalty, which was taken and converted by Scott Sinclair who extended his record of scoring in every league match for Celtic this season.
Tom Rogic rounded things off to make it 6-1 and give Celtic their first win since the 5-1 victory over Rangers.