Kris Boyd's first-half goal proved decisive and consigned Hibs to a relegation play-off and Killie chairman Michael Johnston has thrown his weight behind the beleaguered manager.
Manager Johnston said: "I thought we deserved to win. It was nervy. We were throwing ourselves in front of shots and crosses. You could see how much it meant to them.
"They are always a threat from set-pieces but we defended well. Kris Boyd is a massive player. His experience is invaluable. I hope to start planning for next season."
The former Rangers striker, who has 22 goals this season, has been linked with a return to Ibrox but Johnston will do his utmost to persuade the striker to stay on once he recovers from impending ankle surgery.
"He's a massive player for us," Johnston said. "Give him half a chance, he's going to score goals. We are going to try our very best to keep him.
"You've got to try to keep your best players. I'm sure there will be loads of clubs wanting him but he is a Kilmarnock lad and I know he is happy here. So we can only try our best.
"He needs to get a tidy-up operation on his ankle. He has played through the pain barrier and has been struggling for about a month with his ankle, and it just shows you the character and determination to do well."
Allan Johnston has been under fire from the Rugby Park support this season but his chairman believes he is still the man for the job.
Michael Johnston said: "I'm delighted for the fans who turned out in huge numbers and for the players who have tried so hard. They have shown 100% commitment to the club.
"Allan has another year on his contract, that's the position. We have discussed his plan's for next season. We've stood by Allan in difficult times this season.
"We decided Allan was the man to lead the team. Hibernian chairman Rod Petrie is a tough campaigner and we've learned that you have to put up with the hard times."