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.
The next breakthrough came early in the second set and Medvedev survived two break points in the sixth game, as Fucsovics' best chance at a fightback slipped from his grasp. "Today, I served very well, this is my best serving day so far".
The Frenchman will now face Medvedev, who cruised past Martin Klizan 6-4 6-1.More news: Brad Pitt spotted at Jennifer Aniston's 50th birthday party
Marton couldn't find the way to impose his shots from the baseline and Daniil went 2-1 up in the second set with a forehand return victor in game three, repelling two break points at 3-2 to remain in front. "I knew I had to fight and I'm glad it went well". Saturday's other semi-final sees Matteo Berrettini - a 4-6 7-5 6-4 victor over Fernando Verdasco - take on Marton Fucsovics, who benefitted from 2016 champion Roberto Bautista Agut's withdrawal due to illness.
At the Open Sud de France in Montpellier, 2012 victor Tomas Berdych fought off two match points to secure a 7-6 (7-3) 6-7 (5-7) 7-5 win over Filip Krajinovic. The young Russian lost the opening set against Robin Haase in the second round but he never looked back after that, winning the following eight sets and going all the way for the first success in 2019 after three titles claimed in the previous season.More news: Foot Locker Announces $100M Strategic Investment In GOAT Group
The two players were meeting for the first time on the ATP Tour and made it a contest to remember.
Berdych, who is unseeded at the tournament he won in 2012, then turned the match around.
The Czech's last-four opponent will be seventh seed Pierre-Hugues Herbert, who ousted Denis Shapovalov 7-5 7-6 (7-4).More news: New WWE Fastlane Matches Advertised In Cleveland Market
Tsonga next plays Radu Albot of Macedonia, who won 6-2, 7-6 (7/2) against qualifier Marcos Baghdatis of Cyprus.