The Welsh club had been due to fly from Cardiff Airport at 10am on Wednesday, but fog caused an initial delay to their departure.
A technical fault with their plane then ensured the Swans squad did not take off for Switzerland until 5pm UK time. They eventually landed in Zurich at 8pm local time before going on to St Gallen by coach.
It meant Swansea were unable to train at St Gallen's AFG Arena ahead of Thursday's fixture, where the Premier League side will hope to secure their place in the knockout phase of the Europa League.
Swansea will guarantee a spot in the last 32 if they avoid defeat in Switzerland, while a loss could still see them advance if Kuban Krasnodar do not claim three points at group winners Valencia and overturn a goal difference deficit of three.
Swansea and Kuban have identical head-to-head records after playing out two 1-1 draws in the group, with the Russian club scoring stoppage-time equalisers on both occasions.
While the players were delayed, manager Michael Laudrup was able to attend his pre-match press conference at 6pm local time, having travelled from Paris after meeting family in the French capital on Tuesday.
The Dane admitted the disruption was far from ideal, but will not use it as an excuse should they fail to perform on Thursday evening.
He said: "We cannot control everything, especially the weather.
"It is a very long day for the team. It started at 7.45am at the training ground in Swansea, first they had problems with the fog and then when the plane arrived it had mechanical problems.
"They will arrive about 9pm at the hotel, so when I heard that I cancelled training.
"If it's possible we will do some small things on Thursday just to get them moving.
"Of course, it's not ideal but let's not make it more dramatic than it is.
"It's a very long day for the players and the staff, 12 hours from when they started in Swansea to Zurich.
"It should have been less than half that. The players could have had a rest and then we would have trained and had dinner but they will come, we will have something to eat, and try and get a good night's sleep.
"The game is at 7pm so there will be whole day to prepare for it.
"It's the last game in the group stage and we have to finish the job to move on to the next stage where we want to play in a couple of months' time in the last 32. We have to do the job."
Laudrup confirmed striker Wilfried Bony had travelled as he recovers from an ankle injury, while goalkeeper Michel Vorm was also among the squad despite carrying a knee injury which will ultimately require surgery.
Leon Britton is out with a broken toe, but Michu could play a part having made his comeback from an ankle problem against Hull on Monday, while Angel Rangel is doubtful following a calf strain.
St Gallen have nothing to play for as defeat to Kuban in their last European fixture ensured they cannot reach the knockout stages.
But similarly Valencia have already secured top billing in the group, and Laudrup believes that means Swansea cannot count on a favour from the Spanish club.
He said: "I think we have to do the job ourselves.
"It's true that if Kuban lose or draw we can go through with a loss. But Valencia are already top of the group and are playing Atletico Madrid away on Sunday, which is the top team in Spain, along with Barcelona, right now.
"They know it will be a tough game so I think being already top and having that game in mind in three days' time, I don't think they will play many players against Kuban who will play on Sunday.
"So I think Kuban have a good chance of winning the game there which means we have to do it ourselves."