Swansea qualified for the last 32 of the Europa League despite suffering a 1-0 defeat to St Gallen in their final Group A match.
The Welsh club knew a draw would be enough to guarantee their progression from Group A, 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.
Swansea's preparations for this fixture had been far from ideal after fog and technical issues ensured their journey to Switzerland on Wednesday took over 12 hours.
Some players were also awoken by a shallow earth tremor in the early hours of the morning.
But they managed to take an early control of the match, with Lamah curling just wide and later firing narrowly off-target with a low shot.
The hosts had a goal chalked off for a push on Neil Taylor but found the winner their efforts deserved when Mathys prodded in a low cross from Sebastien Wuthrich.
Swansea still reached the latter stages of a European tournament for the first time in their history, and they will be in Monday's draw for the last 32 along with the likes of Benfica, Ajax, Napoli and Juventus.