Western diplomats, who spoke on condition of anonymity, were sceptical of the claims.
They said anti-government fighters had suffered a large number of casualties and were running low on ammunition, but were not yet defeated.
Gedi said the city should be secure by Friday.
Residents, however, said that clashes on Thursday around an anti-government stronghold in the north were the fiercest yet.
Ethiopian tanks ploughed into northern Mogadishu, firing mortars and rockets on to suspected enemy positions, as machine-gun fire ricochetted across neighbourhoods.
"We are under heavy artillery and tank shelling. The Ethiopians are using whatever forces and material they have," said a fighter belonging to the capital's dominant Hawiye clan.
Anti-government fighters have fired back with machine guns, missiles and rocket-propelled grenades.
"This is the heaviest attack we've seen since the war started," the man said.
One missile slammed through the roof of a nearby children's hospital packed with wounded civilians.
Wilhelm Huber, regional director for the SOS Children's Villages, said the shell that hit the children's hospital exploded in a ward housing between 20 and 30 wounded adults.
The children had been evacuated earlier because shells were hitting the compound, he said.
"What is happening now cannot go on," he said. "People are desperate. This is a tragic situation."
Gedi has urged those who were fighting against the government, and an estimated 340,000 who fled the capital, to return to their homes until his administration could incorporate them into a new national army.
The Fighting in Somalia is being felt as far away as Australia.
Links between a local Somali community and fighters in Mogadishu are currently being investigated.
From Melbourne to Mogadishu
"People can now return to their homes," he said. "The rest of the fighting will be over soon. We have captured the stronghold of the terrorists. We will capture any terrorists who have escaped."
The UN has said the displacement of nearly 340,000 people was turning Mogadishu into a "ghost city" and tens of thousands of refugees were suffering appalling conditions.
John Holmes, a UN emergency relief co-ordinator, warned all sides in the conflict they were breaking humanitarian law by firing indiscriminately on civilian areas.
"All factions are equally guilty of indiscriminate violence in a civilian area," he said.
He urged international donors to quickly meet a $262m UN appeal for Somalia which was so far only 36 per cent met.
Locals and rights activists say at least 300 people have died, and hundreds more been injured, in the most sustained battles since Ethiopian forces ousted the Union of Islamic Courts in a war over the New Year.