Substitute Cedric Itten, playing in front of his home crowd, scored six minutes into his Switzerland debut to give them a 1-0 win over Georgia and leave them with one foot in Euro 2020.
The 22-year-old, who plays for local side St Gallen and missed nearly all of last season with a knee injury, headed in a Denis Zakaria cross in the 77th minute to break down stubborn opponents at the Kybunpark.
Switzerland, second in Group D with 14 points, were left needing a win away to basement side Gibraltar on Monday to qualify.
Leaders Denmark (15) visit Dublin to take on Ireland (12) in their final game on Monday.
Itten's goal came almost immediately after team-mate Renato Steffen escaped a red card for a lunging, studs-up, ankle-high tackle on Georgia's Otar Kakabadze.
Georgia, who completed their eight games with eight points, cannot qualify through the group but are guaranteed a place in the Nations League playoffs which offer four more places at the finals.
They carved out several good chances and Zuriko Davitashvili hit the post in the first half despite being on the ground.