Killie striker Eamonn Brophy fired the visitors ahead in the 33rd minute against the run of play but the Hoops' French hit-man headed in a leveller a minute before the break then slid in a second eight minutes after the restart for his eighth goal of the season.
Ryan Christie added a third soon afterwards before Hoops goalkeeper Fraser Forster saved a late penalty from Rugby Park midfielder Alan Power as the home side kept three Ladbrokes Premiership points ahead of chasing Rangers, who beat St Johnstone 4-0 earlier in the day.
Celtic have now won six out of six league matches and after eight successive Premiership title wins, look more than capable of the ninth, with a performance which – by and large – will have pleased manager Neil Lennon.
After a minute's applause in tribute to former Rangers captain Fernando Ricksen, who died last week after a six-year battle with Motor Neurone Disease, Celtic bossed the start of the game.
The Killie number one made three saves from the Scotland winger and in the 15th minute – when he was beaten by a drive from the edge of the box – the ball smacked the post and flew to safety.
There was even an unsuccessful penalty claim when Power blocked a shot from Forrest.
However, the home side found themselves behind when Brophy took a wonderful pass from Mohamed El Makrini and jinked inside the Hoops penalty area before driving high past Foster, perhaps with the aid of a nick off defender Kristoffer Ajer.
But with a minute of the first-half remaining, Edouard almost-casually rose to power in a header from Forrest's cross to ease increasing anxiety.
Branescu made a great save from Ntcham's curling shot soon after the break but he was beaten again by Edouard after 53 minutes when he slid in from a tight angle after being played in by Bauer.
Celtic were in top gear. Christie turned the ball in from close range three minutes later after Branescu had made a great save from Forrest's header, following a terrific Boli Bolingoli delivery.
Foster stretched to tip a Brophy drive over the crossbar in the 68th minute and – with five minutes remaining, when Killie were awarded a contentious penalty by referee Greg Aitken for Scott Brown's challenge on substitute Osman Sow – he parried Power's spot-kick clear to huge cheers from the Celtic faithful.
There was still time for Forrest to clip the crossbar with a deflected shot as Celtic threatened once more at the death.