In a breathtaking performance, a Kris Commons double and a Gary Hooper strike had the points tied up for the visitors before the interval.
Ireland international striker Anthony Stokes scored a fourth at the start of the second half and in the end the punch-drunk Ayrshire side were happy not to concede any more goals.
The match build-up had been overshadowed by the revelations that parcel bombs had been sent to Celtic boss Neil Lennon.
And there was increased security at the game. Police came out with Lennon and sat behind him in the dugout, and there were also officers in the directors’ box.
The Bhoys managed to keep their heads though. Celtic assistant manager Johan Mjallby praised Celtic’s performance but insists they must target wins in all of their remaining matches this season.
“The pressure is always on and we know in the run-in we need to win more or less every game,” he said.
“We played really well tonight and well done to the players, they turned up here.”
Celtic closed to within a point of leaders Rangers following the win which bolstered their already superior goal difference, and Mjallby praised their away form this season.
“I think we have been strong away from home but Kilmarnock have been really good this season, but in the end we played really well and everyone is focused on the run-in,” he added.
Kris Commons scored twice in the first half and was disappointed not to complete a hat-trick after missing a good chance before getting on the scoresheet.
“It was a good performance, we came out of the blocks nice and quick and in the end it could have been five or six,” he said.
“I should have finished the first one off, so I’m a little disappointed,” Commons added.
Commons also believes Sunday’s Old Firm derby could not have come at a better time as the club are on a fantastic run of form.
“We got a good three points tonight and we are thoroughly looking forward to the game,” he added. “It will be very tough and we know that we have to put a lot of effort in the training sessions during the week.”
KILMARNOCK: Jaakkola, Clancy (Berntsson 79), Pascali, O’Leary, Hamill, Fowler, Kelly (Hay 65), Bryson, Eremenko, Silva, Gros (Agard 83).
CELTIC: Forster, Wilson, Majstorovic, Mulgrew, Izaguirre (Samaras 73), Forrest, Brown, Kayal (Ki 59), Commons (Ljungberg 69), Hooper, Stokes.
Referee: Alan Muir (Scotland).