Zidansek, who will face American Madison Brengle in the second round, won 6-7 (1/7), 7-6 (7/2), 9-7 after three hours and 20 minutes.
Canadaâ€™s Andreescu, the injury-plagued 2019 US Open champion, had been a fitness doubt for the tournament after pulling out of an event in Strasbourg last Wednesday.
But she was forced into a gruelling deciding set against Zidansek, which saw the world number seven fail to serve out the match at 5-4 and then save a match point two games later.
Zidansek created two more match points in the 16th game of the third set and needed only one to wrap up victory.
The longest womenâ€™s singles match in French Open history remains Virginie Buissonâ€™s first-round win over fellow Frenchwoman Noelle van Lottum in 1995, which lasted four hours and seven minutes.