The Czechs were 1-0 down following Tashuma Ito's victory over Radek Stepanek on Sunday but Berdych recorded a 6-1 3-6 6-1 win over Go Soeda to draw level in the Red Group clash.
Stepanek and Frantisek Cermak then triumphed 6-2 7-6(4) in the doubles when seeing off the pair of Soeda and Ito.
"The clay surprised me, after the rain the courts were in pretty good condition," said Berdych. "They were a bit slower but that will always happen."
Philipp Kohlschreiber had beaten Alex Bogomolov Jr. on Sunday.
"Of course, it’s easier when you’re 1-0 up to go on to the court," said Mayer. "But I also knew that I had to win my match, because you never know what happens in the doubles."