The BBC said the portrait of the artist's lover Marie-Therese Walter was expected to sell for between $14 million and $20 million.
The piece was donated anonymously to the University of Sydney, with the stipulation the school use the money from its sale for scientific research.
The university's Dr. Michael Spence told the BBC it was a "very generous and far-sighted gift."
"We are grateful for their extraordinary generosity and delighted with the outcome of the auction," Spence said.
2014: The Year in Music [PHOTOS]