The visitors were struggling to deal with a more lively Celtic side and Georgios Samaras could, and perhaps should, have given the home side the lead after 10 minutes when he rounded Cammy Bell in the Kilmarnock goal but his shot smashed the side netting.
In the dying minutes of the first half Killie thought they had grabbed a shock lead.
Lukasz Zaluska was caught out by a looping ball by teenager Ross Barbour from the left and, as he and Kelvin Wilson tried to clear, Killie claimed the ball crossed the line.
Referee Callum Murray looked to his assistant but waved play on.
Lennon was forced to shuffle his pack after Emillio Izaguirre hobbled off but substitute Ledley fired a rocket past Bell to put Celtic one up.
The former Parkhead striker Sheridan pulled Kilmarnock level just after the interval, though, following a great cross from William Gros.
Five minutes later Matthews put Celtic ahead again with a thunderous shot that rivalled Ledley's for the spectacular and from that point Celtic never looked back.
Gary Hooper, subject of yet another failed bid from English Premier League side Norwich, passed up a number of chances to stretch the lead further but Killie were on the verge of crumbling.
Stokes, on for the injured Samaras, made it three after rounding the goalkeeper and slotting past Bell from a tight angle.
Matthews put the icing on the cake with his second and Celtic's fourth shortly after, tapping in from a yard out.
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