The Premier League side left it late to dispose of opposition operating two divisions lower, with Gary Monk nodding home a 90th-minute winner in their 3-2 success.
Successive defeats by Aston Villa and Everton had left manager Laudrup demanding a reaction and he felt his players answered in the correct manner.
"It's good to get that win. When you lose more than one game it's really important to get back on track and get that feeling of a win," he said.
"To win tonight and do it the way we did, coming back after going 2-1 down, it was good to see that reaction.
"I was worrying that we'd have to go into extra time, which wouldn't have been so good."
Goals from Josh Simpson and Hope Akpan overhauled a 27th-minute opener from Michu, putting Crawley on course for a shock victory.
But visitors Swansea responded through Danny Graham and Monk to ensure an embarrassing evening against npower League One opposition was avoided.
"The more games we have the better it is for everyone because you never know what can happen in the cup," Laudrup said.
"You get through one round, then the second and third. Suddenly you're in a quarter-final.
"Especially for smaller clubs like us, this cup is very important.
"It's the same for smaller clubs like Crawley. It's important to them and that's why the games are tight and sometimes you see some huge surprises."
Crawley's Jonathan Forte made several surging runs from midfield, but his finishing was wayward, letting Swansea off the hook twice in the second half.
Home manager Richie Barker was proud of his side's performance and knew they had gone desperately close to securing a famous scalp.
"I thought at 2-1 the next goal was important. We wanted the daylight and you need that against a Premier League club," he said.
"We have lost out to the odd goal to Manchester United, Stoke and now Swansea. We are getting closer as we are scoring against them.
"The pleasing thing is the performance again."
