This game started as a real scrappy affair and in the first half hour of the match there was very little in the way of pressure from either team.
Dundee did try to up the tempo a bit when, after 33 minutes, a break down the right saw Tom McManus cross to Neil Barrett, but the midfielder shot well wide of the target.
Kilmarnock's Steven Naismith had the game's first shot on target eight minutes before half-time, but from 16 yards his effort was straight at Derek Soutar. Gary McDonald headed Peter Leven's left-foot corner narrowly over the cross bar early in the second half.
Moments later Kilmarnock's Kris Boyd should have done better than fire straight at Derek Soutar from 12 yards Boyd came close again for the visitors just after the hour mark, but his 25-yard free-kick was a bit too straight and Soutar saved easily.
After 62 minutes Dundee were awarded a penalty when David Lilley fell Neil Barrett just inside the box. Steve Lowell stepped up to take the kick and fired high into the net past Alan Combe.
Boyd went close again 20 minutes from time when he headed Danny Invincibile's right-wing cross just over the bar. Moments later Steve Lowell spotted Combe off his line and his speculative 40-yard effort went just past the post.
Tom McManus had a good right-foot shot well saved by Combe 11 minutes from time, and moments later Kilmarnock captain Lilley saw his six-yard header hit the cross bar and go over.
With five minutes to go the visitors almost grabbed an equaliser but the home defence did their job twice in a matter of seconds when firstly Soutar saved to deny Craig Dargo, and when the ball eventually fell to Steven Murray his fierce drive was headed clear by Bobby Mann one yard from the goal line.