Herbert breezed through the opening set but Berdych offered more resistance in the second, only for the former world number four to crack at a key moment.
A Berdych double fault at 5-5 in the second set earned Herbert a break and the Frenchman then served it out to claim his first win against the Czech.
He will face either compatriot Jo-Wilfried Tsonga or Moldova's Radu Albot on Sunday.
(Reporting by Julien Pretot; Editing by Pritha Sarkar)
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