Colin Doyle admits Kilmarnock are relieved they have not been left with a mountain to climb to rescue their Premiership place.
It was an error-strewn display from the Ayrshiremen but they finally got the break of the ball with 13 minutes left as Brandon Haunstrup's shot broke back to him off Dees striker Danny Mullen, with the Killie full-back then grasping a precious lifeline at the second attempt.
However, goalkeeper Doyle had to pull off two great saves to ensure it was not game over by the time Haunstrup pounced.
First he bailed out the Englishman after his horror fresh-air swipe sent Paul McMullen through on goal, while he made another vital stop to deny Mullen after the break.
But the Irishman is just relieved Kilmarnock are still in with a chance of extending their 28-year top-flight run when they host James McPake's side in Monday night's Rugby Park return.
"We are still in the tie," he said. "It is only one goal and we have another 90 minutes.
"We are at home and we need to be confident to see if we can turn it around.
"There is no point in beating around the bush, this wasn't the performance or result we wanted but we are still in the tie.
"It could have been worse, it could have been 3-0 or whatever. We are still in it and 90 minutes is a long time in football.
"We are confident and we will also have our fans back as well so we are looking forward to that. We have to be positive."
Wright has masterminded a huge turnaround in form since being appointed back in February.
Thursday's defeat was just the second Killie had suffered inside 90 minutes in their last 10 games but they slipped up as Dundee whipped up a storm with goals from Jordan McGhee and Charlie Adam.
Doyle said: "The conditions were terrible. It was blowing and the wind was swirling, it was coming down with rain, it is not nice for goalkeepers or anyone.
"We have managed to get back in the game. We will see what happens at home.
"We were nine unbeaten apart from the Motherwell game. We were in good form but that can happen in football.
"It is frustrating because we wanted the win and worst case a draw. I have been in football a long time and I have seen these situations overturned.
"It is going to be difficult because they will be confident. We have to manage that and to try and get an early goal to put a bit of pressure on them.
"We will be confident and hopefully we can celebrate on the astro on Monday.
"We want to win and stay in the league and we will give everything to get the result to keep the team in the Premiership and to keep the fans happy."