Hoops striker Odsonne Edouard fired the visitors into the lead in the 25th minute before strike partner Leigh Griffiths added a second in the 50th minute with a drive.
The Parkhead side looked to be cruising before Killie striker Nicke Kabamba reduced the deficit with a header in the 66th minute but a Christopher Jullien header from a corner seven minutes later clinched the victory.
Celtic remain two points ahead of the Light Blues, who beat St Mirren 1-0 at Ibrox and have a game in hand.
The start to the match was certainly lively with Celtic showing early intent.
Griffiths, partnering Edouard in a 3-5-2 formation, slipped on the artificial surface when he broke into Kilmarnock's penalty area in the fourth minute, with Edouard also unsteady seconds later.
Then, returning defender Jozo Simunovic's powerful header from a Griffiths corner was brilliantly clawed away from the line by Jan Koprivec, and then the Rugby Park keeper made a good save from the Scotland striker's 25-yard free-kick.
However, in between, the 31-year-old Slovenia cap, making only his second Killie appearance, badly misjudged an aerial challenge with Griffiths inside the box and the ball broke to Edouard, who uncharacteristically missed the unguarded goal with a shot.
Callum McGregor had an equally good opportunity when burrowed his way into the Killie box after taking a pass from the French forward but he too somehow missed the target from 14 yards.
However, when Edouard got a second chance he made no mistake.
Celtic wing-back Jeremie Frimpong skipped past Niko Hamalainen down the right and his cross into the box was trapped by the 22-year-old attacker, who skillfully rolled Killie midfielder Alan Power before driving the ball between the legs of Koprivec from eight yards.
Alex Dyer's side wobbled under further Parkhead pressure but held out before edging back into the game, though the Hoops were worth their interval lead.
Kabamba, making his first start after signing from Hartlepool, watched his curling shot from the edge of the box at the start of the second half deflect past for a corner, but that came to nothing.
However, moments later Koprivec did not look too clever as he allowed Griffiths' low drive from a McGregor pass beat him at his near post, Frimpong again involved in the move.
Killie's task now appeared enormous.
Koprivec saved from Edouard in the 62nd minute after the Parkead striker had eased past defender Stuart Findlay, who was perhaps lucky his opponent chose to stay on his feet inside the box after his challenge.
Celtic looked in complete control but when Kamamba redirected a Power shot past Forster, the home side took heart.
The visitors kept their heads, however, and Jullien restored their two-goal lead when he powered in a header from a Griffiths corner and with that, the points were Parkhead bound.
There will be concern that Frimpong was taken from the field a couple of minutes from the end on a stretcher after a challenge by Power, replaced by Moritz Bauer.