In its latest update at around 1500 GMT, the Mauritius Meteorological Services (MMS) agency said Freddy had "passed at its closest distance from the island" at about 120 kilometers to the north and was moving west-southwest at a speed of about 30 kilometers an hour.
The MMS said that although Freddy had undergone "a slight weakening", thunderstorms and gusty weather would continue.
"Sea will be high with heavy swells of the order of seven metres beyond the reefs. Storm surge will continue to cause inundation along the low-lying coastal areas. It is, therefore, strictly advised not to go at sea."
"The public in Mauritius is advised to maintain all precautions and to stay in safe places."
Images from the remote paradise island showed waves crashing to the shore and the wind whipping through palm trees.
In an address late Sunday, Prime Minister Pravind Jugnauth urged citizens to take all necessary precautions, stay home and remain "vigilant."
"Cyclone Freddy is an extremely strong cyclone which is a direct threat" to the islands of Mauritius, Rodrigues and Saint-Brandon, he said.
The MMS had earlier lowered the cyclone's classification to intense from very intense and lifted its safety bulletin for the autonomous island of Rodrigues which lies 600 kilometers east of Mauritius.
As citizens holed up indoors, police said security services remained on alert to tackle any emergencies.
Airports of Mauritius announced that the international airport would be closed from Monday until further notice "due to the cyclonic weather".
"Air Mauritius is closely monitoring the situation with the authorities and will keep passengers informed of developments," the national carrier added on its website.
The idyllic holiday destination is renowned for its spectacular white sandy beaches and turquoise waters but also lies in the pathway of occasional cyclones.
The authorities on the French island of Reunion, which is expecting the cyclone to reach it overnight Monday, have also gone on alert.
- 'Rising seas' -
The island nation of Madagascar was also bracing for Freddy, which is expected to make landfall on Tuesday night, according to the country's National Risk Management Office (BNGRC).
The agency warned of heavy rainfall, "strong or stormy" seas and wind gusts with a speed of 250 kilometers per hour, in a statement published on its Facebook page.
"There is a serious concern about the rise of the sea, especially along the area where it makes landfall" along the eastern coast, it said.
"After making landfall, Freddy will cross the center of the country during the night and is expected to exit the Mozambique Channel in the southwest region on Wednesday afternoon," it said.
The agency warned citizens to stay away from the sea "from tonight because the weather conditions will worsen due to the Freddy-generated waves".
About a dozen storms or cyclones occur each year in the southwest Indian Ocean during the November-April season.
Some information in this report came from Agence France-Presse.