Already in uncharted territory in her first quarterfinal on grass, the 18-year-old French Open finalist had to fight back from 2-0 down at the start of each set against an opponent who was the Wimbledon runner-up last year.
Gauff said she needed some time to find her rhythm and found the answer by mixing up her shots to include more slices.
"It was really tough to be honest. I had to come up with a couple different game styles that I normally don't use, but sometimes it's like that," she said.
Pliskova missed out while trying to reach her first final since August. The Czech player missed the first two months of this season when she broke her arm in an accident while training in the gym.
Fourth-ranked Jabeur won against Aliaksandra Sasnovich 6-7 (3), 6-2, 6-2 to reach the semifinals.
Kasatkina dropped the first set in her previous two matches in Berlin this week before comeback wins but couldn't find a way back against Sakkari's power hitting.
"I'm very pleased that I had the right game plan. I went for it, I was very aggressive, and I really enjoyed it," said Sakkari, who has won three straight matches for the first time since March and her run to the Indian Wells final.