Zverev took just 77 minutes to ease to a 6-2 6-4 victory over Prizmic, who was ranked more than 100 places below the new US Open and French Open champion.
Just like in Saturday's victory over Matej Dodig, the second-ranked Zverev dominated on the indoor hardcourt in Halle, Germany, with Prizmic having no break-point chances.
"The Davis Cup here at home in front of the home fans was incredible," said Zverev.
"That is exactly what the Davis Cup is all about for me. We have a strong team that deserves to be at the Finals."
Earlier on Sunday, Kevin Krawietz and Tim Puetz overcame Nikola Mektic and Mate Pavic 7-5, 6-2 to hand Zverev the opportunity to send Germany to the Final 8 in Bologna, Italy, in November.
"That was probably the best doubles match we've ever played in Davis Cup. Everything clicked from start to finish, and the atmosphere was outstanding. We were delighted to secure this important point," said Krawietz."
Germany's Davis Cup captain Michael Kohlmann added: "Qualifying for the finals for the third time in a row is very special and by no means a given.
"We put in a very strong team performance here. The atmosphere was outstanding on both days, and we are now looking forward to the finals."
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.
Great 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.
And doubles pair Henry Patten and Neal Skupski beat Gonzalo Escobar and Diego Hidalgo 6-3 6-4 to secure a 3-0 win for the hosts in London.
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