The Gunners exited this season's competition on Wednesday night, going out on away goals despite a 2-0 win over Bayern Munich in the second leg of their last-16 tie at the Allianz Arena.
That leaves Arsene Wenger's men needing to finish in the top four if they are to feature in the continent's premier club competition next term.
The north London club currently sit fifth in the table, five points adrift of fourth-placed Chelsea and seven off Tottenham in third, although they have a game in hand on Andre Villas-Boas' men.
But Laudrup believes they cannot afford give up any more ground when they take on the Swans at the Liberty Stadium this weekend.
He said: "They are disappointed because they are out of Europe, but I think maybe they had accepted that after the first game against Bayern at the Emirates.
"But they come out of this game better than they went in because they have won away against a very good side who are one of the favourites for the Champions League.
"They are still five points behind fourth place in the Premier League and I think they will now be going for that, and they have to believe until the very end.
"They will try to keep this confidence, Arsene Wenger knows after Saturday's game there is an international break, so it is very important for them.
"Chelsea are at home this weekend, so if they win and Arsenal don't win here, they would be eight points behind and with only nine games to go that may be too big a gap.
"So it is a very important game for them to keep alive the possibility of going into the Champions League next season."
Laudrup will again be without defender Chico Flores and midfielder Kemy Agustien against the Gunners as they recover from their respective ankle and thigh injuries.
But Leon Britton should be available to play some part after returning to training following the hip injury he suffered at West Brom.
There was also good news on left-back Neil Taylor, who will make his return from the broken and dislocated ankle which has sidelined him since September.
The Wales international was due to play 45 minutes for the Swansea Under-21 side in their game at QPR on Thursday.
Laudrup said: "He is playing 45 minutes today [Thursday] so it is a great day for him.
"H has been looking well in training, he has a way to go to get his match fitness back but that will come with games.
"It's good for him and it is nice to have him back.
"There is a psychological barrier after being out for so long but he is doing well."
Swansea have also confirmed midfielder Mark Gower has joined Charlton on loan until the end of the npower Championship season.
Gower has found opportunities limited this season, making just three first-team appearances.