In the men's side of the draw, No 1 seed Casper Ruud crashed out in straight sets to French teenager Arthur Fils.
Fils burst out of the blocks, winning the first set 6-0 and the second 6-4, withstanding a fightback from the world No 4.
Fils credited his "good tactics" for the win.
"He makes you run a lot. My goal was to play a lot of forehands and play a lot on his backhand, and I did it really well today."
The 19-year-old will meet Alexander Zverev in the semis.
World number 19 Zverev, who like Akugue was born and raised in Hamburg, defeated France's Luca Van Assche, 6-3, 6-4, to make the last four of the tournament for the third time.
Serbian Laslo Djere defeated titleholder Lorenzo Musetti of Italy, setting up a meeting with China's Zhang Zhizhen.
World number 57 Djere won in straight sets 7-5, 6-3 against the 2022 Hamburg Open winner and third seed.
The 28-year-old, who has won two previous titles on clay, will face Zhang, who defeated Daniel Altmaier.
Zhang, ranked 79 in the world, won 6-4, 6-4 after breaking the German once in each set on the way to his first ATP semifinals.