First-half goals from Paul Heffernan and Dean Shiels proved enough to take Kilmarnock past First Division Dundee with a 2-1 win in their William Hill Scottish Cup fourth-round replay at Rugby Park.
Killie overcame off some early lethargy before striker Heffernan opened the scoring in the 34th minute with Shiels doubling that lead two minutes from the break with a fine strike.
Dundee's goal came in the 63rd minute when Killie midfielder Liam Kelly failed to prevent Dundee counterpart Gavin Rae reaching Gary Irvine's cross and the ball sped past Cammy Bell to bring the game back to the boil.
However, the Ayrshire side held out to book a place against Hibernian at Easter Road in the next round although they will probably have to improve on this performance if they are to progress any further in this competition.
Killie boss Kenny Shiels claimed Dundee were the favourites going into the game, and those who believed it was a case of him playing mind games would have thought again in the early stages as Dundee were more than a match for their Clydesdale Bank Premier League opponents.
However, Killie's opener was basic. Bell thundered a kick down the pitch, Gary Harkins headed on and Heffernan ran onto the ball before drilling a left-footed shot low past the diving Robert Douglas from inside the box for his 12th goal of the season.
The hosts' second goal owed everything to the splendid finishing of Shiels. After a Dundee move broke down in their own half, Harkins set up the Northern Irishman who flicked the ball up and drove high past Douglas from the edge of the box.
Dundee chased and harried the Ayrshire side who looked uneasy again - trying, but mostly failing, to get their passing game going.
And just after the hour mark the visitors were back in the game. Rae went up for the header with Kelly, and the Dundee player was soon celebrating with the travelling fans behind the goal after the ball ended up behind Bell.
The play swung from end to end as the Dark Blues went all out for the equaliser but an increasingly nervy Killie side, in the end, held out without too many goalmouth scares.