It was the top-seeded Swiatek's first match on clay since she won the 2023 French Open and took her career record to 64 wins and just nine losses on the surface.
"First match on clay (for the year), so it's always tricky," she said after her one hour 33 minute victory.
"Elise was really good. Sometimes at defence I had to be patient and really work for the points. So I'm happy with the way I handled it.
"That's one of the things you have to switch from hard court, you have to be ready for longer rallies and for players to be able to get back more."
Raducanu, the 2021 US Open champion who last week helped Britain beat France to qualify for November's Billie Jean King Cup finals, knocked out Linda Noskova 6-0 7-5.
"She seems like she's getting her game back together, so it's great to see that, because for sure she has potential," Swiatek - chasing a hat-trick of title wins in Stuttgart - said of her last-eight rival.
"It's not going to be easy. She knows how to spin the ball and how to move on clay, as well."
Fourth-seeded Elena Rybakina battled her way into the quarter-finals with a roller-coaster 7-6 (7-3) 1-6 6-4 victory over Veronika Kudermetova.
Rybakina staved off three break points towards the end of the opening set and then took control of the tiebreaker, but her errors mounted in the second set as Kudermetova began to dominate.
The 2023 Australian Open finalist regained control in the decider and surged to 4-1 before more errors allowed Kudermetova to level.
But the world No.4 rallied again to take a tight service hold and close out the match.
She will face Jasmine Paolini after the Italian upset seventh-seeded Ons Jabeur 7-6 (8-6) 6-4.
Sixth-seeded Marketa Vondrousova, the Wimbledon champion, also reached the last eight, seeing off Anastasia Potapova 7-6 (7-5) 6-1.
She next faces back-to-back Australian Open winner and world No.2 Aryna Sabalenka.
Ukraine's Marta Kostyuk will come up against Coco Gauff in the quarter-finals after knocking out fifth seed Qinwen Zheng of China 6-2 4-6 7-5.
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