Portuguese midfielder David Silva's 81st minute goal from close range proved to be the difference as Killie, who had skipper Garry Hay sent off late in the match, took their unbeaten run to three games, allowing them to go clear of fifth-placed Inverness CT whose midweek clash with Dundee United was postponed.
The loss, their second in a matter of days, dropped 'Well to 8th on the table and manager Stuart McCall was flattened by the result as it compounded the misery of going down in the Co-operative Insurance Cup semi-final to Rangers on the weekend.
"I thought Sunday was bad but this was a real sore one to take," he said.
"I find it really difficult to believe we lost the game.
"I have only seen Kilmarnock once and people were telling me how good some of their players are - and they are - but we snuffed out any threat they had.
"Our commitment and desire was good but the end result was a real sore one to take.
"I don't think anyone who was at the game would say that we deserved to lose the game.
"Even a point would have been a disappointment if I'm honest."