Swiatek forced her way back but a series of double faults, including on the final point of the set, allowed the American to claim the opener.
The 25-year-old found her feet to claim the second set, dropping just two games, but Navarro fought back and broke Swiatek to claim the third set and the match in two hours and five minutes.
“It was hot out there and I didn’t have a ton of rhythm, I think both of us felt that throughout the match. So it came down to a lot of serving,” Navarro said.
“I relied on my serve there at the end and it ended up getting me through.”
In the quarterfinals, Navarro will face Romanian Elena-Gabriela Ruse, who beat Russian Anna Kalinskaya 7-5, 6-2.
Earlier on Wednesday, French Open champion Andreeva fell in her first match since her Roland Garros victory, losing 6-3, 6-4 to Alexandrova.
World No. 5 Andreeva, playing her first match on grass in 2026, pulled out before last week’s Berlin Open on the surface, saying she wanted more time to recover and prepare for Wimbledon.
The 19-year-old matched her compatriot early in the opening set before losing seven straight games in hot conditions in central Germany.