The two teams have experienced contrasting fortunes since winning their respective opening games, and St Gallen will be playing merely for pride on Thursday.
Jeff Saibene's team enjoyed a 2-0 triumph over Kuban Krasnodar on matchday one but four consecutive defeats have followed.
Nevertheless, the Swiss team have previous when it comes to upsetting Premier League opposition.
In 2000 they stunned Chelsea in the first round of the UEFA Cup, overturning a 1-0 first-leg loss at Stamford Bridge with a famous 2-0 home triumph thanks to goals from Sascha Muller and Charles Amoah.
The current crop were also on the receiving end of a 1-0 scoreline in the UK when Wayne Routledge struck to settle October's meeting at the Liberty Stadium.
That victory followed up Swansea's famous 3-0 win in Valencia, but revenge for the Spanish side in Wales last time out on the back of two draws against Kuban ensures Michael Laudrup's men still have work to do.
A St Gallen win at AFG Arena combined with victory for Kuban over Valencia and a three-goal swing in the Russian outfit's favour would stop Swansea taking a European campaign into the new year for the first time in their history.
Laudrup is boosted by the availability of Michu and the talismanic forward will be looking to get back on the goal trail having returned from ankle and knee problems in Monday night's 1-1 draw with Hull.
A knee injury to Michel Vorm means Gerhard Tremmel is set to continue in goal for City and the German stopper is confident ahead of the tie.
"Qualification is in our hands, and that's very important," Tremmel told Swansea's official website.
"Of course, it also depends on how Kuban do in Valencia, but we will only be concentrating on our match.
"We want to qualify for the next round by getting a good result in St Gallen.
"Of course, it won't be easy, as proved when we played them at the Liberty. But the whole squad is determined to go to Switzerland and get the result we need to get through."
Forwards Daniel Sikorski and Dzengis Cavusevic remain long-term knee injury absentees for St Gallen, while defender Mario Mutsch must serve a one-match suspension.
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1