"Our country doesn't have the skills and expertise to refloat stranded ships, so I have appealed for help from France and President Emmanuel Macron," he said. Bad weather had made it impossible to act and "I worry what could happen Sunday when the weather deteriorates".
Jugnauth shared a photo of the vessel, the MV Wakashio, tilted precariously.
Video posted online showed oily waters lapping at the shore as people murmured and peered at the ship in the distance. Online ship trackers showed the Panama-flagged bulk carrier had been en route from China to Brazil.
The French island of Reunion is the closest neighbour to Mauritius, and France's Foreign Ministry says France is Mauritius's "leading foreign investor" and one of its largest trading partners.
"We are in a situation of environmental crisis," the environment minister of Mauritius, Kavy Ramano, said, calling the Blue Bay Marine Park and other areas near the leaking ship "very sensitive."
After the cracks in the hull were detected, a salvage team that had been working on the ship was evacuated, Ramano told reporters on Thursday. Some 400 sea booms have been deployed in an effort to contain the spill.
Government statements this week said the ship ran aground on July 25 and the National Coast Guard received no distress call. The ship's owners were listed as the Japanese companies Okiyo Maritime Corporation and Nagashiki Shipping Co Ltd.
A police inquiry had been opened into issues such as possible negligence, a government statement said.
Tons of diesel and oil were now leaking into the water, environmental group Greenpeace Africa's climate and energy manager Happy Khambule said in a statement.