Herbert breezed through the opening set but Berdych offered more resistance in the second, only for the former world No. 4 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.
Wildcard Tsonga beat Moldova's Radu Albot 6-1, 6-3 after overcoming an early break in the second set.
Doubles specialist Herbert, who won the Australian Open title with Nicolas Mahut last month, will be looking to claim his first singles title on Sunday.
Tsonga will be bidding for his 17th singles title but his first since 2017.