The champions remain top of the SPL as goals from Scott Brown, Joe Ledley and Georgios Samaras sees the Champions League last 16 participants continue their good away form
The victory sees the Glasgow side remain at the top of the SPL and continues their fine away form in the league which has saw them score 24 goals on their travels, compared to seven at home.
The home side made only one change from the side that defeated Queen of the South in the fourth round of the Scottish Cup, with Scotland cap Liam Kelly coming back into the side in place of Lee Johnson.
Celtic made four changes from the team that qualified for the last 16 of the Champions League, with Mikael Lustig, Kris Commons, Beram Kayal and Samaras making way for Ledley, Adam Matthews, Victor Wanyama and Lassad Nouioui.
In the opening stages Gary Hooper, Brown and Wanyama all tried their luck at goal, but none of them could find the target and a way past Cammy Bell in the Kilmarnock goal.
Kilmarnock rarely ventured forward, but when they did, former Celtic striker Cillian Sheridan managed to get in behind the visitors defence, before lobbing the ball towards the back post, were Paul Heffernan was, but he could not make anything of the chance.
The home side then found themselves behind, after some slack defending allowed Hooper to find Matthews racing down the right wing, he cut the ball back to Brown in the centre of the box and the Hoops skipper placed the ball beyond the helpless Bell.
Kenny Shiels' side should have got level a minute after the break as Jeroen Tesselaar broke down the left-hand side, before crossing for Heffernan, but the striker took too long with his effort and that allowed Mulgrew to block the strike.
Hooper then had two chances in the space of a couple of minutes to add to Celtic’s lead, but on both occasions the Hoops striker found Bell in good form as he managed to divert each effort away from goal.
Kilmarnock started to see a lot more of the ball, but they could not make anything of their chances and were ultimately punished as Wanyama played the ball from the edge of the area to Brown on the right-hand side and his back post cross was met by the onrushing Ledley, who bundled the ball home.
Things got even worse for the home side as Celtic kept possession, passing the ball about with freedom and that allowed Brown to find Matthews in space on the right-hand side and the full-back raced forward before whipping a cross in for substitute Samaras to head home from close range.
As the game entered injury time, Kilmarnock managed to get a goal back as Sheridan got onto a cross from the left, but it was too little, too late for the Ayrshire side.