The 13th-ranked American, who knocked out defending champion Coco Gauff in the fourth round, triumphed 6-2, 7-5 after trailing 5-1 in the second set.
She will face either second-ranked Aryna Sabalenka or Olympic champion Qinwen Zheng for a place in the final.
"When I got to 5-2, I had an inkling that I'd win in two sets," said Navarro who had lost in the first round on her only other two appearances at the tournament.
"Semi-finals baby. I'm ready to rock."
Navarro swept through a 29-minute first set with breaks in the second and eighth games as a tense Badosa was undone by 16 unforced errors to the meagre five of her opponent.
The New York-born Spaniard hit back and raced into a 5-1 lead in the second before her game fell apart with Navarro taking the last six games of the match.
Badosa, who was on the brink of retirement due to a back injury just three months ago, finished the quarter-final plagued by 35 unforced errors. Navarro had just 15.
If Zheng defeats Sabalenka in her quarter-final later on Tuesday, it will set up an intriguing showdown after Navarro blasted the Chinese star following her defeat at the Olympics.
Navarro accused Zheng of being a "cut-throat" and of showing a "lack of respect".
"They are both big hitters. They will come after me but I'll be ready," said the American.
