Sharapova, 31, will play top seed and world number one Simona Halep in the semis after the Romanian crushed Caroline Garcia 6-2 6-3.
Sharapova produced one of her best performances of the season to overcome her Latvian opponent in a gripping contest lasting three hours and 10 minutes.
Fifth seed Ostapenko saved a set point before edging the opener in a tiebreak but failed to maintain her level in the second as Sharapova forced a decider.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
The Russian missed two match points at 5-3, allowing Ostapenko to draw level after 10 games, but Sharapova was not to be denied as she booked a semi-final spot in Rome for the first time in three years.
"It's great to back at this stage," Sharapova said.
"My fans have been so loyal throughout the years and for me to be able to produce this kind of tennis again in front of them is very special."
Estonia's Anett Kontaveit knocked out world number two Caroline Wozniacki 6-3 6-1 to record her second consecutive win over a top-10 opponent.
Kontaveit, who overcame Venus Williams in the last 16, struck 23 winners and broke her Danish opponent's serve six times to wrap up victory in just over an hour.
The world number 26 set up a semi-final battle with defending champion Elina Svitolina of Ukraine, who defeated former world number one Angelique Kerber 6-4 6-4.
(Reporting by Hardik Vyas in Bengaluru, editing by Ed Osmond)
