The Hoops' 2-1 home win over Spartak Moscow on Wednesday night sealed their place in the last 16 of the Champions League and they continued their good work in Ayrshire, starting when skipper Scott Brown fired them into the lead in the 27th minute.
Second-half goals from midfielder Joe Ledley and substitute Georgios Samaras, on for Lassad Nouioui, were just reward for the Parkhead side's dominance, before former Hoop Cillian Sheridan grabbed a late consolation.
Celtic have struggled at times in post-European games this season and indeed had taken only one point from nine following their previous three Champions League games, including a 2-0 home defeat to Killie in October.
However, there was no such danger this time as Neil Lennon's side cemented their place at the top of the SPL.
Brown's goal to break the deadlock was one of simplicity. When Adam Matthews took a pass from Gary Hooper and picked out the Scotland midfielder from the byline, he guided the ball quite calmly past Cammy Bell from 12 yards.
Celtic stretched their lead through Ledley when the home side failed to properly clear their lines.
Victor Wanyama picked up the loose ball, played in Brown and when his driven cross came into the middle the sliding Wales international forced it over the line from a few yards out.
Samaras made an immediate impact coming off the bench. Following another driving run by Matthews, the big Greek striker headed the Wales international's deep cross past Bell and in off the bar.
Efe Ambrose should have made it 4-0 a minute from time when Charlie Mulgrew planted a free-kick on his head but the Hoops defender headed wide and there was enough time left for Sheridan to volley a Jeroen Tesselaar cross past Fraser Forster.