West Ham midfielder Tomas Soucek had a penalty saved by Switzerland goalkeeper Yann Sommer
Switzerland only needed a draw to stay in League A and led 2-0 thanks to goals from Remo Freuler and Breel Embolo.
But the Czechs battled hard as West Ham's Vladimir Coufal hit a post before Patrik Schick pulled a goal back.
Another West Ham player, Tomas Soucek, had a penalty saved by Switzerland's Yann Sommer and Jan Kuchta also hit the woodwork for the visitors.
The win means Switzerland finished third in Group A2, with Spain winning the section and advancing to the four-nation finals tournament next June thanks to a dramatic 1-0 away win over Portugal.
Former Stoke City and Liverpool winger Xherdan Shaqiri (centre) featured in Switzerland's win over Czech Republic
The Czech Republic join Austria, England and Wales in dropping down to League B in 2024-25.
-- Courtesy of BBC Sport