The Colombian, ranked 66th, put up a formidable fight after double-faulting to lose a testy first-set tie-break, though she needed medicine to push through trouble with her back.
Osorio saved two match points but Gauff, the world number four, dug deep, rescuing long rallies with a solid backhand and cross-court saves.
"It was tough, it wasn't the cleanest," said the 22-year-old American after the two-hour-and-18-minute match.
"Just happy with how I was able to reset."
She next faces either Spain's Cristina Bucsa or Chinese teenager Sun Xinran, world number 582.
Later on Saturday, world number one Elena Rybakina is scheduled to play Armenia's 118th-ranked Alina Charaeva in the night session.
In the men's draw, top seed Alexander Zverev will play home hope Shang Juncheng for a chance to face Novak Djokovic in the quarter-final.