Twenty people have been injured, according to officials in Vilanculos.
Roads were swamped with rain and blocked by uprooted trees, and power supply was disrupted.
Experts say that once the cyclone gets further inland it will begin to lose much of its power, although heavy rainfall is likely to continue even as the wind speeds drop.
Zambezi valley threat
The National Meteorology Institute said Favio's storms were concentrated in the province of Inhambane but were felt as far away as Xai-Xai, the capital of nearby Gaza province.
The storm was moving north towards the central Zambezi River valley, which is already struggling with severe flooding after weeks of heavy rains.
There are growing fears that Favio would bring more misery to flood-ravaged central Mozambique, where tens of thousands of people live in temporary shelters with scant food or water.
Officials said the problems could multiply in the coming days as Favio dumps its rains in Zambia, Malawi and Zimbabwe, further swelling the tributaries that feed the Zambezi.
Radio Mozambique said that the cyclone has caused widespread damage at the holiday resort of Tofo Beach, uprooting palm trees and destroying electric pylons.
Mozambique's worst disaster in recent history occurred in 2000-2001 when a series of cyclones compounded widespread flooding in southern and central parts of the country, killing 700 people and driving close to half a million from their homes.
Metrorologists say they are also tracking another cyclone, Gamede, which is growing in the Indian Ocean east of Madagascar, but as yet there are no indications of whether it will make landfall.