The game, which just passed a late pitch inspection after a torrential downpour in Kilmarnock, was delayed for 45 minutes because of a floodlight problem.
Once the game did get started it was Celtic who got into their stride dominating possession, but Kilmarnock reduced them to very few chances in the first half.
Bulgarian midfielder Stilian Petrov, who was rested for the midweek win over Hearts, missed a great opportunity to put The Bhoys in front when he headed over Jackie McNamarra's right-wing cross from six yards out unmarked over the crossbar.
The closest the home side came to breaking the deadlock in a scrappy first half was when Ally Mitchell's corner flicked off the head of Bobo Balde towards goal, but the Alan Thompson was there clear the ball off the line.
Celtic came out all guns firing in the second half as they tried to make the crucial breakthrough and they found it just five minutes after the restart with an unfortunate own goal from Gary Hay.
Thompson whipped in a dangerous corner into the near post towards Henrik Larsson and although the prolific Swede missed the ball it deflected off Hay's foot, past the helpless Gordon Marshall for the opening goal of the game.
The visitors took control of the game and they doubled their lead on the hour mark and it was Larsson who netted the goal.
The Swede ran onto Thompson's delightful lofted pass over the Kilmarnock back line to coolly slot home left-footed for his 24th goal of the season.
The impressive Thompson almost got his name on the score sheet in the final few minutes with a superb chip which sailed just inches wide.
Killie had a chance to pull a goal back in stoppage time when, former Celtic striker Tommy Johnson forced Rab Douglas into a fine save, but the following up Kris Boyd could only steer his shot wide with the goal gaping.
That meant that Celtic repeated the 2-0 success they earned at Rugby Park just two weeks ago.
Martin O'Neill's men will now face Aberdeen in the next round and they will be out to avenge their only defeat of the season at Pittodrie after suffering a 2-0 defeat to The Dons in December.
Click on fixture for full match summary
Kilmarnock 0-2 Celtic
