In 2008, Mirzakhani joined the faculty of Stanford University, where she served as a professor of mathematics until her death.
The 40-year-old reportedly had breast cancer that spread to her bones.
She was survived by her husband Jan Vondrák, a Czech theoretical computer scientist and her colleague at Stanford University and a daughter named Anahita.
Mirzakhani enjoyed pure mathematics because of its "elegance and longevity", she said.
Born on May 3, 1977, in Tehran, Mirzakhani showed her knack for numbers from an early age, winning back-to-back gold medals in the 1994 and 1995 International Mathematical Olympiads. She was the first female recipient since the award's establishment in 1936.
"Maryam is gone far too soon, but her impact will live on for the thousands of women she inspired to pursue math and science", Stanford President Marc Tessier-Lavigne said in a statement.
Iranian-American scientist Firouz Naderi shared his grief on Instagram.
When she won in 2014, the IMU called Mirzakhani's accomplishments in complex geometric forms such as Riemann surfaces and moduli spaces "stunning".
"A light was turned off today, it breaks my heart...."
"A genius? Yes. But also a daughter, a mother and a wife", he added in a subsequent post. In Iran, President Hassan Rouhani said that Mirzakhani's "doleful passing" has caused "great sorrow", state media reported.
Iranian Foreign Minister Mohammad Javad Zarif paid tribute to her.
Mirzakhani completed her PhD at Harvard in 2004, then accepted positions as a research fellow at the Clay Mathematics Institute and an assistant professor at Princeton, accruing awards and acclaim along the way.
Stanford University has not responded to a request for comment from Newsweek.
Maryam Mirzakhani is awarded with the 2014 Fields medal at the International Congress of Mathematicians in Seoul, South Korea on August 13, 2014.