Kilmarnock secured a welcome place in the fourth round thanks to two goals in the closing minutes of the first half.
Motherwell adapted to the dreadful conditions, both overhead and underfoot, better than their opponents throughout the match, but could not find a way past Kilmarnock captain David Lilley.
Former Kilmarnock favourite Gordon Marshall produced the first real save of the game in the 34th minute when Danny Invincibile set up Kris Boyd, and the young striker unleashed a powerful drive which the keeper did well to parry.
Kilmarnock took the lead three minutes before the break when Motherwell failed to clear their lines. Steven Naismith's corner fell into the six-yard box and sparked a monumental scramble which ended with Gary MacDonald stabbing the ball home.
Kilmarnock scored a killer second on the stroke of half-time. Again Motherwell were found wanting defending a corner, this time Boyd rising to meet Naismith's cross unopposed and heading the ball home, despite Kevin McBride's efforts on the goal-line.
Motherwell refused to lie down and kept pushing forward, but left gaps in their defence which Kilmarnock looked likely to expose. Invincibile went close before Garry Hay and Naismith got in each other's way when a third goal looked likely.
Motherwell's afternoon was summed up in the 79th minute when their skipper Scott Leitch blasted beyond Alan Combe from 30 yards only to see his effort cannon back from the post.
Richie Foran was lucky not to be sent off when his frustration got the better of him and he flared up at Combe, but Stephen Craigan was not so fortunate and he was ordered off in the closing minutes for his second yellow card.