At least eight of the victims were caught in underground areas as the flood waters rose rapidly. Another died of a heart attack.
The island's metereologists said 152mm (6in) of rain fell in less than an hour, 70mm less than the March average.
Prime Minister Navin Rangoolam declared 1 April a day of mourning.
Speaking on national radio, he said Mauritius was suffering badly from the effects of climate change.
The floods caused chaos in the city, with a huge traffic jam paralysing its centre, L'Express de Mauritius newspaper reported.
A BBC reporter in the city says the bodies of six people were recovered from a pedestrian subway and another two from an underground car park.
More rain is expected on Sunday, and people have been advised to stay at home. -BBC