Killie worked tirelessly and gave Celtic a hard shift, but never looked likely to breakdown the giant visiting defence.
The home side's new loan signing Liam Fontaine was thrown in at the deep end by Jim Jefferies, handed the job of marking Sutton, and the youngster gave an impressive display which augurs well for the remainder of his spell at the club.
There was a breakdown in communication in the home defence in the fifth minute when no-one challenged for Ulrik Laursen's cross leaving Alan Combe exposed, but Bobo Balde could not mark his final Celtic appearance with a goal as the ball clipped the bar and went behind.
Sutton was the main danger for Kilmarnock and he had two good chances to score in the first quarter.
First he headed Alan Thompson's corner narrowly over before the winger engineered an even better chance in the 20th minute.
Thompson spun away from Gary Locke on halfway, spotted Sutton's run and played the perfect defence-splitting pass, but the ex-Chelsea man hit his shot weakly at the keeper.
The home fans thought they had taken a surprise lead when Steven Naismith swept home at the far post, but Gary McDonald had failed to keep the ball in play before he crossed.
Stilian Petrov enjoyed a storming display for Celtic, and almost scored with a powerful shot which was only inches wide.
Sutton then headed Aiden McGeady's in-swinging corner just past as, for all Kilmarnock's resistance, a goal seemed inevitable.
The breakthrough came in the 37th minute via the penalty spot. Naismith had instigated a running feud with Neil Lennon and gave away a free-kick midway inside his own half after an unnecessary nibble at the Irishman's ankle.
Thompson floated the ball into the danger area where it struck Danny Invincibile's hand. The referee was in no doubt that it was intentional and pointed to the spot.
Combe guessed correctly but Sutton's strike was true and found the bottom left corner of the goal.
The second half petered out as Celtic were held in check by Killie's determination, but the home side never looked like equalising.
John Hartson's frustrating afternoon was summed up in the 88th minute. He was penalised regularly for being overly physical - challenges he would normally be allowed - and when given a gilt-edged opportunity to wrap up the match after Thompson's perfect pass along the six-yard line, he lifted the ball over Combe but onto the face of the crossbar.
To Kilmarnock's credit, the final whistle was greeted with roars of relief from the travelling support.