The Russian eighth seed needed just under an hour on Court Philippe Chatrier to reach the last four at Roland Garros for the second time in three years.
The crushing win brought to an end a fairytale run for 36-year-old Cirstea, who will retire at the end of the season. The in-form Romanian reached the quarter-finals in Paris for the first time since 2009.
"I knew that this match wouldn't be easy and I'd have to put 200% of my focus to win," Andreeva said.
"I'm super happy I was able to do it and tried to play aggressive throughout the whole match."
Andreeva will next face the winner of the all-Ukrainian last-eight match-up between Elina Svitolina and Marta Kostyuk.
