They improved to eventually take control and led at the break thanks to goals by Maciej Zurawski and Jiri Jarosik.
Shunsuke Nakamura added a third with a fabulous free-kick in the second period before Steven Naismith scored a late consolation for Kilmarnock. There was still time for Zurawski to add his second and Celtic's fourth in stoppage time to make the victory more emphatic for Gordon Strachan's side.
Celtic's style has changed dramatically under Strachan who prefers to play the ball on the ground and rely on skill rather than power.
The home team should have taken an early lead only for Kenny Miller to fire in a weak shot at the back post. Killie defender Fraser Wright then did well to block a Zurawski drive when the striker found space in the box.
The home team survived a scare when Gary Caldwell was dispossessed by Naismith but Bhoys keeper Artur Boruc did well to block the winger's shot.
Celtic went ahead in the 25th minute when Miller latched on to a poor first touch by David Lilley before curling in a superb cross for Zurawski to side-foot home.
After Naismith had missed a great chance for the visitors, a flowing Celtic move ended with Jarosik heading home Zurawski's cross from ten yards out.
Kilmarnock were simply swept aside in the second half as Celtic's one-touch football left the visitors chasing shadows. Caldwell saw a glancing header cleared off the line by Garry Hay and the only surprise was that the hosts had just a superb Nakamura free-kick, curled into the corner of the net from 28 yards, to show for their domination until late on.
In the last three minutes Naismith volleyed home for Killie after Boruc appeared to have been blocked. The Poland international goalkeeper was booked for his protests.
Zurawski made the scoreline more realistic when he swept home Aiden McGeady's mis-hit effort in stoppage time.