Wayne Routledge got the only goal of the game six minutes into the second half to give the home side the win, tapping in at the far post.
Swansea were a long way short of the form they showed in humbling Valencia at the Mestalla in their first group game, and the visitors would have taken something from the match had striker Goran Karanovic converted a first-half penalty.
The game was hardly helped by the inconsistent and fussy refereeing of Duarte Gomes, but Swansea can now head into the home game with Kuban Krasnodar later this month knowing a win would put them a long way towards a place in the last-32.
Into the bargain, Michu secured his status as a cult hero after suffering a cut to his head courtesy of the boot of Stephane Besle in the first half.
Michu got to his feet with blood streaming from a cut and a farce developed as efforts to stem the flow, including vaseline, bandages and a change of shirt, all failed.
He was then prevented from returning to the field as blood seeped onto his change shirt, before a numberless jersey was finally deemed up to standard and he wandered back on.
At half-time a member of Swansea's staff was sent to the club shop to retrieve a shirt with his name and number on the back, but even that was changed for a fifth of the night during the closing stages.
"We talk so much about him." said Laudrup.
"But it shows he just wants to play, he is very important for the other players when he is in there.
"He works so hard and he is really important for us, but I guess that's not really news."
He added: "We did not know how bad it was, because he was on the other side of the field.
"We said he would come back on, but we were eight minutes with 10 men.
"Fortunately nothing happened, it was not bad for him and we did not concede.
"There was a little bit of blood, so he changed to one with no number.
"He had some stitches at half-time. I don't know how many."
When asked about the dash to the club shop for a new shirt, Laudrup joked: "Wasn't it closed?"
While Michu managed to remain on the field, St Gallen's Karanovic was forced off at half-time after a heavy collision with Swans' keeper Gerhard Tremmel. The concussion he sustained meant he had no memory of his miss from the spot.
Laudrup, meanwhile, was satisfied to see his side in such a strong position after two games.
"We are in a fantastic position, Valencia won away in Russia so we have six points, and there are two teams on three points and they play each other next time," he said.
"If we could win the next game it would be a really good position."