On home soil, the No.1 seed won 80 per cent of total service points in the opening set but Bonaventure fought back in the second to send the match to a deciding set.
A valiant effort from Kontaveit in the third set secured a spot in the final four, where she will face compatriot Kaia Kanepi after her 6-2 3-6 7-6 (8-6) victory over Karolina Muchova in the final match of the day.
Earlier in the day's play, No.2 seed Belinda Bencic defeated Donna Vekic to set up a semi-final clash against Barbora Krejcikova, who defeated Beatriz Haddad Maia in straight sets.
At the Parma Ladies Open, the scheduled semi-finals were postponed because of rain and will instead be played on Sunday (AEDT), with the final to be played later in the day.
No.1 seed Maria Sakkari faces Danka Kovinic, with play to begin at 10:30am local time, and will be followed by Ana Bogdan against Mayar Sherif.