Wayne Routledge scored the only goal of the game as Swansea made it two wins from two matches in the competition this season, moving three points clear at the top of Group A in the process.
Britton admitted that they didn't start the match well, with St Gallen missing a penalty inside 15 minutes, but was happy at having picked up another win.
"The first 25 mins it was a bit sloppy and with the penalty - maybe we could have had a penalty ourselves," Britton told ITV Sport.
"It wasn't a great performance. It was pretty gritty but it's two from two."
Swansea will now turn their attention back to domestic action as they face Southampton in the Premier League on Sunday.