American Navarro was trailing 14 in the opening set against 113th-ranked Saville before reeling off five straight games to take control of the match.
She won 6-4 6-2 and set up a semi-final against Britain's Katie Boulter, who beat 2022 runner-up Donna Vekic 6-4 6-3.
"Someone in the crowd told me to wake up. I guess that was maybe the push I needed," said 22-year-old Navarro, who capitalised on eight of her 14 break-point opportunities.
"It's tough playing someone I've never played before. I was just kind of getting used to her game and putting together a strategy of how I wanted to play.
"It took me a little bit to find that, but I got there in the end."
It will be Navarro's fifth tour-level semifinal in less than a year. She won her first WTA title earlier this year at Hobart, Australia.
World No.49 Boulter had ousted second-seeded Beatriz Haddad Maia 3-6 6-3 6-4 earlier this week.
Top-seed Jessica Pegula beat Anna Blinkova 6-1 2-6 6-2 in their quarter-final and the world No.5 will face sixth seed Marta Kostyuk who downed fourth seed Anastasia Pavlyuchenkova 3-6 6-4 6-3..
Pegula finished with 24 winners. She made 17 unforced errors, the majority of which came in the second set.
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