A lower back injury led to No.6 seed Milos Raonic, who was taking part in his first tournament since the Miami Open in March, withdrawing from the all-Canadian final-four match in Germany.
"I'm very sorry and apologetic that I'm unable to compete. I was hoping for the best after my match yesterday," the 2016 Wimbledon finalist Raonic said.
"I woke up this morning not feeling my best. I wanted to come out here and give it a try. And after looking at it very diligently with my team, and the way I was feeling with my back, I found it was not possible for me to compete without taking any greater risk."
Berrettini faced his first two break points of the tournament but staved them off en route to a 6-4 7-5 victory over home hope Jan-Lennard Struff.
In Den Bosch, Borna Coric saved two match points but fell a mini-break behind in a third-set tie-break before his semi-final against Adrian Mannarino was interrupted by rain.
Jordan Thompson claimed an important break to move 7-5 5-3 up against Richard Gasquet but was facing a break point in game nine of the second set when play was suspended.
The players will conclude their matches on Monday (AEST) before the final is contested.