The Welsh club needed a draw to guarantee their progression, but Kuban Krasnodar's failure to beat Valencia in Spain ensured this defeat did not prove costly.
In sub-zero temperatures and with a low fog hindering visibility, the visitors had the better of the opening half with Roland Lamah twice going close.
But they slackened off after the break and were punished when Marco Mathys found the net from close range with 10 minutes to go.
It was an inauspicious way to end a group campaign that had started with a 3-0 Mestalla masterclass against Valencia, but Swansea will be in Monday's draw along with the likes of Benfica, Ajax, Napoli and Juventus.
And while the result and performance were not what would have been wanted in Switzerland as Swansea limped over the line, Laudrup was happy to have progressed.
"Being through is the most important thing," he said. "In a couple of weeks we won't think about how we got there, we will just know we are in the last 32.
"We have two months to wait before the knockout stage, and we are there because we deserve to be there.
"Maybe you can always take things out from these games and say we should have done more, but we are through.
"When we were 1-0 down and Kuban equalised against Valencia, for just a second their late goals against us went past my mind.
"When St Gallen got a last-minute free kick - two goals in injury time could have sent us out, but fortunately it didn't go like that."
Kuban's 1-1 draw in Spain means Swansea have reached the latter stages of a major European tournament for the first time in their history, and Laudrup hopes it is an achievement the club will be able to look back on fondly in years to come.
"I hope that it is huge for everybody in Swansea. We don't know when we will play in Europe again," he said.
"Sometimes you appreciate it but maybe not enough when you are in it. Maybe you appreciate it more when you are more distant from it.
"I can only compare it with my experience at Getafe when we reached the quarter-finals. It was a nice experience at the time, but four or five years later is when they really appreciated playing Bayern Munich and Benfica.
"I hope thinking back it will be the same with Swansea, when it comes to winning the League Cup and playing in Europe.
"Now let's see what happens in the draw. You never know, the weaker team of the two can move on if you play well."
Laudrup also said it would be a couple of days before the club know the full extent of the knee injury suffered by defender Chico Flores, who limped off during the second half.
"I talked to the doctor and we have to wait a couple of days to know. It was a knock to his knee, but we can't say any more about how long he will be out," he said.
"He was already suspended for Norwich on Sunday, but we will have to wait a couple of days to know how bad it is."
Source: PA
Source: PA