Kilmarnock won all three points, leaving the home team ruing numerous missed chances.
With eight minutes played Kilmarnock took the lead when a shot by Kris Boyd was blocked by Lee Wilkie, but fell nicely for Colin Nish who had the easy task of slotting the ball into the net from eight yards Midway through the first half Nacho Novo ran at the visitors defence and had a shot from 25 yards but his shot was straight at the keeper and was easily saved.
Mark Fotheringham for Dundee fired inches wide for the home team just before the half hour mark and a minute later Peter Canero should have had the visitors two up, but screwed his shot across the goal from ten yards.
With 32 minutes on the clock Dundee were awarded a penalty when Novo was brought down by Francois Dubourdeau.
The little Spaniard took the kick himself and made no mistake sending the keeper the wrong way.
Five minutes later Killie were back in front when Boyd beat Lee Mair in the air to head Garry Hay's cross past Julian Speroni from eight yards.
Novo almost equalised for Dundee a second time eight minutes after the break, but shot narrowly wide after running past Frederic Dindeleux.
Novo went close again with an hour played but this time saw his well-struck shot carried away by Dubourdeau.
Dundee were seeing most of the ball, but failed to make it count and despite 16 goal attempts Dubourdeau in the visitors goal was rarely being tested.
With seven minutes remaining a Dundee corner was floated into the box at the second attempt and Tom Hutchinson headed goalwards, but his effort was again easily saved by Dubourdeau.
With two minutes of time added on Dundee's midweek hero youngster Bobby Linn was handed a great chance to be ahero again, but when standing all alone six yards out he headed Garry Brady's cross wide of target.