Hoops skipper Scott Brown, returning from a two-game suspension, headed the reigning Scottish Premiership champions ahead in the 29th minute.
Edouard converted a penalty in the 51st minute and the French striker added his second 10 minutes later with an accomplished finish before fellow forward Albian Ajeti added a fourth with three minutes remaining.
Boss Neil Lennon had asked for a response from his players following the damaging 2-1 home defeat to St Mirren on Saturday which he admitted was a new low for him.
There was clear improvement although whether that is enough to appease the Northern Irishman's critics is another thing as the Parkhead men languish 20 points behind league leaders Rangers with a game in hand.
Killie's head of football operations James Fowler and assistant Andy Millen were in the home dugout after Alex Dyer had left following the defeat by St Johnstone at the weekend and he watched the rudderless Ayrshire side slump to a fourth successive defeat.
There was no surprise that Lennon rang the changes with a debut for right-back Jonjoe Kenny, who arrived on loan from Everton on deadline day with 35-year-old Brown, defender Stephen Welsh, midfielder Ryan Christie and striker Ajeti all back.
The home side were asked to defend studiously in the early stages as Celtic came at them from left, right and centre and there was an ignored penalty claim when Edouard went down in a packed penalty box.
Killie almost took the lead, however, in the 11th minute when a terrific cross from Rory McKenzie found Chris Burke in the six-yard box but under pressure from former Killie left-back Greg Taylor, he seemed to get the ball stuck under his feet and danger was averted.
Celtic resumed control and again Killie retreated.
In the 19th minute midfielder David Turnbull's effort from 25 yards just escaped Doyle's left-hand post.
However, the visitors' pressure paid off after Ajeti won a corner on the right and Turnbull's delivery saw Brown escape marker Alan Power and head the ball past Doyle from 10 yards for his first goal in a year.
Doyle saved a long-distance drive from Turnbull just before the interval as Celtic remained on top.
Edouard had the ball in the Killie net early in the second half but Christie was well offside although he was soon to get on the scoresheet.
Kevin Clancy pointed to the spot after Ajeti tumbled to the ground when challenged by Doyle and Edouard slotted the spot-kick only just past the Killie keeper, who was far from convinced about his guilt.
Killie stepped it up and a close-range Burke header from a McKenzie cross was blocked on the line by Celtic keeper Scott Bain.
That was as close as Killie got to getting back into the game.
In the 61st minute Edouard ran onto a Turnbull pass – there was a hint of offside – and knocked the ball past the advancing Doyle.
Ajeti rifled in a fourth from inside the box in the closing minutes after taking a Brown pass, with time left for Bain to make a decent save from Power's angled drive.