It made landfall on Thursday with winds of up to 177 km/h (106 mph), but aid teams only reached Beira on Sunday.
People have been rescued from trees, homes have been destroyed and roofs were ripped off concrete buildings, head of the Red Cross assessment team, Jamie LeSeur, told the BBC.
The cyclone has killed at least 150 people across southern Africa.
More than 80 people have died in eastern and southern Zimbabwe, information ministry head Nick Mangwana told Reuters news agency.
This includes two pupils from the St Charles Lwanga boarding school in the district of Chimanimani, who died after their dormitory was hit when rocks swept down a mountain.
Malawi has also been badly hit. The flooding there, caused by the rains before the cyclone made landfall, led to at least 122 deaths, Reliefweb reports.
How bad is the damage in Beira?
At least 68 people have died in Mozambique, mostly around Beira, the authorities there say.
More than 1,500 people have been injured by falling trees and debris from buildings including zinc roofing, the BBC’s Jose Tembe in the capital, Maputo, quotes officials as saying.
“Almost everything has been affected by the calamity,” Alberto Mondlane, the governor of Sofala province, which includes Beira, said on Sunday.
“We have people currently suffering, some on top of trees and are badly in need of help.”
Local people in Beira, with a population of 500,000, have put in an “incredible effort” to reopen roads in the city, Mr LeSeur told the BBC’s Newsday programme.
“Beira has been severely battered. But we are hearing that the situation outside the city could be even worse,” a statement from the International Federation of Red Cross and Red Crescent Society quotes him as saying.
The road linking Beira to the rest of the country has been damaged, but air links have now resumed.
President Filipe Nyusi cut short a trip to eSwatini, formerly known as Swaziland, to visit the affected areas.
What’s the situation in Zimbabwe?
A state of disaster has been declared in Zimbabwe. President Emmerson Mnangagwa has returned home early from a trip to the United Arab Emirates to “make sure he is involved directly with the national response”, the authorities say.
The ministry of information has shared pictures of pupils from St Charles Lwanga School, who have now been rescued.
‘Frustration is growing’
Shingai Nyoka, BBC Africa, Mutare, eastern Zimbabwe
It’s been days since the first rains fell but the skies here show no signs of clearing.
Roads and bridges have been washed away and air rescue is the only hope for the hundreds who remain stuck.
The low-hanging clouds and mist are stopping rescue efforts and frustration is growing at the command centre here in Mutare.
Social Welfare Minister Sekesai Nzenza told me that helicopters have waited for days to airlift the injured and deliver food and blankets to those affected by the flooding.