The home side had a positive start with a few half-chances, while the Welsh team looked to Wayne Routledge in the early stages of the opening half, but his cross was easily cleared out of the penalty area.
Roland Lamah had the Premier League outfit's first effort of the game by launching a curling shot towards goal from the edge of the box, but the ball just drifted wide of the post.
Michael Laudrup's men eventually displayed some passing play on the field, which allowed Jonjo Shelvey to sprint forward and unleash a shot on target, but his effort could not muster enough power and the ball flew straight into St Gallen goalkeeper Marcel Herzog.
Lamah tried once again to break the deadlock after surging towards the box and thumping a shot 20 yards from goal with around 10 minutes left to play, but his shooting was off target once again as the ball dragged wide of the post.
Swansea need just one point to advance into the last 32.