The Swans recovered from an awful first-half display in Saturday's fourth-round tie at Birmingham to secure their spot in the fifth round.
Lee Novak gave the Blues a deserved lead, but substitute Wilfried Bony's quick-fire double turned the game around to give the visitors a 2-1 win.
Swansea now face a critical league fixture against Fulham on Tuesday where they will look to end a run of eight top-flight games without a win and extend their three-point gap over the bottom three.
Manager Laudrup appreciates that some will view Swansea's FA Cup and Europa League commitments as a distraction, but he is happy to see his squad challenging on as many fronts as possible.
"This result gives us confidence, even if this is another competition, which might be seen as a downside by some," he said.
"In our case we are still in three competitions; is that good or bad?
"I think it is a good thing as it gives your players games and experience.
"I understand some people think it would be better to concentrate on one - the Premier League - but I still believe it is always better to win games than to lose no matter what competition.
"I'm very happy. It would have been very bad in our situation to have had to face a replay.
"That would have meant five consecutive midweek games for us with Premier League games and our Europa League games against Napoli coming up. But we recovered well to get the win."
The win came at the end of a week when Swansea's unity has been questioned following a training ground row between Chico Flores and Garry Monk.
During the first half against Birmingham there was also the sight of captain Ashley Williams letting striker Alvaro Vazquez know what he made of his performance in no uncertain terms.
Laudrup does not doubt his squad's togetherness, but has identified leadership as a key element during the final months of the season.
"It is about leadership, you always need that in a team in every moment, you need it most when you are behind or not doing well," he said.
"It is easy to be a leader at 3-0 up and talk a lot on the pitch, but that is still an area we can improve a lot, to help each other with one or two words.
"If you can say things to help another player see things that he perhaps can't. We have to help each other.
"In this situation things can get worse before they get better and in the first half here it was very difficult but we came through it and now we need to win on Tuesday."
Birmingham boss Lee Clark was disappointed to see his side miss out on claiming Swansea's scalp for the second time this season, having knocked them out of the Capital One Cup.
But he was delighted by the performances of youngsters Reece Brown, fresh from signing his first professional contract, and Albert Rusnak, making his debut after joining on loan from Manchester City.
He said: "Reece is an outstanding young player and we are delighted to have signed him. He is the best footballer at the club technically.
"Rusnak is in on loan, he is very talented and he is right alongside Reece, they are on the same wavelength in terms of passing and movement."