Chiudinelli and Lammer parlayed a pair of breaks into a 5-1 lead in the fourth set. Zimonjic held his serve, but by then it was too late as Lammer closed out the 3 hour match.
Roger Federer and Australian Open champion Stan Wawrinka got the Swiss off on the right foot Friday with singles victories. Switzerland will meet the winner of the Belgium-Kazakhstan series in April.
Serbia took the court without ace Novak Djokovic while Federer bolstered the Swiss squad with his first opening-round Davis Cup action since 2005.
Celebrity Families of 2014 [PHOTOS]