Swansea will travel to Russia in the wake of their south Wales derby defeat at the hands of fierce rivals Cardiff, a performance Laudrup himself labelled "disappointing", while captain Ashley Williams questioned the team's desire.
The long trip east may not appear the ideal chance to turn matters around, but the 1-1 draw between the teams in Swansea two weeks ago ended the Welsh side's 100 per cent record in Group A and means, despite being top of the table, they still have plenty of work to do if they want to secure a place in the knockout stages.
Laudrup does not doubt his side's character, but has made it clear Thursday evening at Kuban Stadium and Sunday's Premier League meeting with Stoke are the times to show it.
"I've said a lot of times the team has character and they will show it," he said. "But it is always good to get confirmation of something and I am sure we will see a reaction from the team because they are so disappointed at this loss.
"The game on Thursday is a five-hour flight away and four hours' time difference. It's a lot of travelling.
"But I've said it before and I will say again, the reason for that is that we deserve it, we won the League Cup last year so we are in Europe. We will try to go to Russia and win because we have some history with them after they scored in the last seconds against us in Swansea.
"Then we face another strong physical team in Stoke at home on Sunday which, just before the international break, is a game we really want and need to win."
Sunday's game at Cardiff saw Laudrup again opt to leave record signing Wilfried Bony on the bench.
The £12million Ivory Coast international has been used sparingly in the Premier League since his summer arrival from Vitesse Arnhem, and Laudrup has revealed that is because Bony is still playing catch-up after having a shortened pre-season.
"When Bony came here he hadn't had a proper pre-season and then after we signed him he had to stay away for nine days until he had his work permit," the Dane said.
"He started playing and in some games he did well, especially the European games and a couple of others. But we want him to be at his best because I'm the first one to know the expectations everyone has of him (because of his transfer fee).
"We are here to help him and that's what we are trying to do, to give him a programme where we are doing extra things with him to get him at his best.
"I think he understands that maybe that means in some games he will not be there. Of course everybody wants to play all the games, but what we are doing is trying to put him up to his best condition so we can see the best of Wilfried.
"I don't know how long it will take but he needs to adapt as it is a big step from the Dutch league to the Premier League."