Cyclone Idai, which is carrying heavy rains and winds of up to 170 km/h (106 mph), made landfall at the port city of Beira on Thursday evening.
Its 500,000 residents are now without electricity and communications have been severed, the National Institute of Disaster Management (INGC) says.
The cyclone is expected to move west to Malawi and Zimbabwe over the weekend.
It follows a week of storms and heavy rains in Mozambique and Malawi that have killed more than 100 people and destroyed thousands of homes.
Beira is the fourth-largest city in Mozambique and its port sits on the mouth of the Pungwe river, that runs to Zimbabwe.
How bad is the damage?
Images posted on social media show significant damage to Beira, with makeshift roofs blown away and trees uprooted.
"Houses and trees were destroyed and pylons downed," the INGC told AFP.
Five people have been injured, according to local media.
The coastal town of Chinde, 400 km (250 miles) north-east of Beira was also badly hit, the INGC says. Nearby villages have been cut off from the mainland by a two-metre tidal surge, according to Reuters news agency.
"Twenty tonnes of biscuits will arrive in country shortly for immediate assistance to stranded communities, by boat and helicopter," said Michael Milton of the UN World Food Programme.
All domestic flights were cancelled on Thursday as Mozambique braced for the storm.
Mozambique has been struck by severe cyclones in the past, including Eline in 2000, when 350 people died and 650,000 were displaced across the wider region.
Beira has often seen the worst of the storms and has worked to limit the effects of rising waters brought about by climate change through a series of infrastructure projects.
The Global Facility for Disaster Reduction and Recovery, a group linked to the World Bank and the UN, says Mozambique is the third most at-risk country in Africa when it comes to extreme weather.