The big-serving Struff upset fourth-seeded Stefanos Tsitsipas 7-6 (7-5) 5-7 6-3 on Thursday to become the third lucky loser to reach the semi-finals of a Masters 1000 tournament.
The 33-year-old German had lost 6-4 6-2 to Karatsev in qualifying but made it to the main draw after another player dropped out.
Struff had lost only five service games in the main draw and relinquished only one against Tsitsipas on his way to victory in the Spanish capital.
"It feels amazing. It was a very, very hard battle. I knew before I needed to perform at my best," Struff said. "Very, very happy that I played this well today."
The fifth-ranked Tsitsipas was coming off a final appearance at the Barcelona Open, where he lost to Carlos Alcaraz. He was a runner-up to Novak Djokovic in Madrid in 2019.
Defending champion Alcaraz faces 17th-seeded Borna Coric in the other semi.
Karatsev reached his first Masters 1000 semi-final after ending a strong run by Chinese player Zhang Zhizhen 7-6 (7-3) 6-4.
The 121st-ranked Russian saved three break points in the first set and converted his only opportunity in the second.
"Aslan is playing amazing this week so far and he beat me pretty easily in qualies, I need to say," Struff said.
"I didn't play the best tennis in that match, but he made me play not good, I feel like. I think we need to analyse it now, focus on the match tomorrow and I hope I can do better."
Karatsev has been ranked as high as 14th. In 2021, he became the first man in his grand slam debut to reach the Australian Open semi-finals.
He became only the second qualifier to make the semi-finals in Madrid.
"I started the year inside the top 100 then I dropped and lost some matches," he said. "You have to keep going and believe, and I have a team behind me that is always supporting me and believing in me."
