Dusan Lajovic and Filip Krajinovic won their so-called dead rubber singles matches to complete the first-round series on the shore end of a 3-2 score.
Switzerland clinched its trip to the quarterfinals against Kazakhstan with a doubles victory against the Serbs on Saturday.
Serbia took on the Swiss without its ace, Novak Djokovic. Lajovic was the highest-ranked player on the Serbian squad at 102nd in the world.
Lajovic topped Michael Lammar Sunday 6-3, 3-6, 6-4. Krajinovic followed with a 6-4, 6-4 blanking of Marco Chuidinelli.