Great Britain ensured they will be one of the eight nations to compete for the trophy over six days in Bologna in November as they comfortably beat Ecuador 4-0.
Austria, Canada, South Korea and the Czech Republic all moved through as they won their qualifiers on Saturday.
Italy qualified automatically as reigning champions.
Zverev got the ball rolling for Germany on Saturday with a straight-sets victory over Matej Dodig but Prizmic levelled the tie by beating Daniel Altmaier.
Kevin Krawietz and Tim Puetz won Sunday's doubles for Germany, leaving Zverev to wrap up the tie with a composed 6-2 6-4 win against Prizmic.
Spain were without Carlos Alcaraz but that did not matter as they beat Chile 3-0 with fast-rising Rafael Jodar impressing on his debut. Jodar thrashed Cristian Garin on Saturday after Daniel Merida had edged out Alejandro Tabilo.
Sunday's doubles saw Pedro Martinez and Jaume Munar breeze past Tabilo and Tomas Barrios Vera in quick time.
Britain were without the likes of Jack Draper and Cameron Norrie but still had far too much for Ecuador as Toby Samuel and surprise Wimbledon semi-finalist Arthur Fery won their Saturday singles against Andres Andrade and Alvaro Guillen Meza.
Henry Patten and Neal Skupski won Sunday's doubles at the Copper Box in London to send the hosts through.