The second-tier side won Thursday's first leg 2-1 at Dens Park and will now be looking to complete the job against their top-flight opponents in East Ayrshire.
Kilmarnock have spent the past 28 years in the Scottish Premiership, but that is under real threat following their defeat in the opening 90 minutes of this finely-poised tie.
However, Brandon Haunstrup's rifled finish 13 minutes from time ensures that Killie are still very much alive with at least 90 more minutes to play.
That strike from Haunstrup was Kilmarnock's only shot on target of the match, so Tommy Wright will know that a vast improvement is needed on Monday.
Indeed, opposite number James McPake will perhaps look back on the first leg and wonder how his side are not in a more commanding position after dominating in front of 5,000 fans.
That win means Dundee have now won four of their last five matches and have lost just one of their last nine, while Kilmarnock have won just one of their last five.
The hosts will fancy their chances of scoring at least once on Monday, though, as Dundee conceded 40 goals in the regular Championship season - only Alloa Athletic and Queen of the South shipped more goals.
Killie have lost only one of their last seven games at Rugby Park, but the Dees have been very strong on their travels, winning four and drawing two of their last six.
More impressive than that, McPake's charges have conceded only one goal across that period. Another clean sheet on Monday and they will be back in the top flight following a two-year absence.
Six of the last 13 meetings between the sides since February 2016 have finished all square, with Killie winning four and Dundee prevailing three times.
Kilmarnock are still without striker Nicke Kabamba, who has been ruled out for the past month with a hamstring injury.
Midfielder Youssouf Mulumbu will also play no part, meanwhile, having made an abrupt exit from the club supposedly over a lack of first-team appearances.
Kyle Lafferty did not enjoy the finest of first legs, but he has 12 goals in 11 appearances since joining in February, making him the club's leading goalscorer this term.
As for Dundee, they lost Christie Elliott to a knee injury in the first leg and the 29-year-old will have to be assessed ahead of this match.
Kilmarnock possible starting lineup:
Legzdins; Elliott, Ashcroft, Fontaine, Kerr; McGhee, Byrne; McGowan, Adam, McMullan; Mullen
Dundee possible starting lineup:
Doyle; Millen, Broadfoot, Rossi, Haunstrup; Power, Dicker; Burke, Kiltie, Pinnock; Lafferty
We say: Kilmarnock 3-1 Dundee (4-3 on aggregate)
Dundee could have been out of sight against Kilmarnock in the first leg, but as it is they have only a one-goal lead to take into this return match.
Killie will be desperate to preserve their top-flight status for a 29th year, and we can see them producing an improved display to turn the tie around on Monday.