The home side had the best of the first period with striker Allan Russell clipping the bar with a shot.
However, the tide turned in the 63rd minute when Keane opened the scoring from Edson Braafheid's long ball and two goals within a minute secured the visitors a place in Monday's semi-final draw.
Like last week's win at Falkirk, where Keane helped himself to a brace, the Republic of Ireland international showed why he is worth his £65,000 per week wages with his eighth goal in eight games since arriving from Tottenham in January.
And having dropped 13 points behind SPL leaders Rangers, Hoops' boss Tony Mowbray will be glad that his side recovered from an indifferent first half in which a cup exit looked possible.
A far from full Rugby Park witnessed a scrappy start to the match.
Celtic captain Scott Brown, playing wide on the right, had the first effort on goal in the fifth minute when he drove at the Killie defence before firing wide from the edge of the box.
Kilmarnock slowly grew in confidence and Russell got the break of the ball in the 14th minute after captain Craig Bryson had driven at the Parkhead defence and his curled shot from 16 yards beat goalkeeper Lukasz Zaluska but clipped the top of the bar.
At the other end a Braafheid free-kick from 25 yards was held by Kilmarnock goalkeeper Cameron Bell, who made an even better save when he blocked Keane's close-range shot after he had been threaded in by Marc Crosas.
But Celtic looked rather disjointed with Keane and Morten Rasmussen isolated up front and the home side pressed with purpose, with only the final pass missing.
In the 38th minute Kilmarnock defender Frazer Wright did well to head away Aiden McGeady's cross from under the bar.
At the other end, Bryson then brilliantly burrowed his way past Braafheid and into the Celtic box, but Russell failed to get on the end of his cut-back.
The visitors, apparently rejuvenated by Mowbray's half-time pep-talk, almost broke the deadlock two minutes after the restart.
Killie failed to clear their lines from a Celtic attack, allowing the foraging Keane to chip the ball up for a McGeady volley which was speeding past Bell and towards the top corner until Wright bravely headed clear.
Moments later, Bell palmed a half-volley from McGeady over the bar for another corner, which again the home side survived.
In the 55th minute Russell again hit the woodwork when Tim Clancy's cross from the left was flicked on by Burchill.
As the Parkhead defence hesitated Russell stole in at the near post but Zaluska did enough to prevent the Killie striker getting a clean shot away and the ball hit the post before bouncing wide.
That was the cue for Mowbray to replace Rasmussen with Georgios Samaras and Crosas with Marc-Antoine Fortune.
But in the 63rd minute it was Keane who gave Celtic the lead and it came courtesy of a long and hopeful ball from Braafheid.
The pass caught the Kilmarnock defence sleeping and Keane's first touch took the ball past Bell and he calmly stroked the ball into the empty net.
As Celtic tightened their grip, Bell made another decent save from McGeady's low drive.
However, the tie was all over in the 81st minute when Samaras robbed the hesitant Wright on the touchline and played in Fortune who rolled the ball past Bell for Keane to tap into an empty net.
A minute later Fortune set up Keane again and this time the Hoops' striker almost casually rattled a drive past Bell to take collection of the match ball.