Residents left the capital on foot, by car or in donkey carts amid fears that more clashes would follow over the coming days.
"Ethiopian tanks are still stationed inside our neighbourhoods and the insurgents are likely to launch counterattacks, so we are leaving for our own safety," Faduma Ahmed, who was fleeing with her six children and her brother, said.'Trapped in the battle'
Up to 68,000 people are believed to have fled since the beginning of the year as Islamist fighters and local clans have staged near daily attacks on transitional government forces and Ethiopian troops.
Somalia has not had an effective government since Mohammed Siad Barre was overthrown in 1991.
The Islamic Courts Union seized Mogadishu from the control of warlords in June 2006.
The transitional government, along with Ethiopian troops, defeated the Islamic Courts in late 2006.
About 6,500 people were killed in violence in Mogadishu in 2007.
The 250,000 civilians camped outside Mogadishu are considered to be the world's biggest group of internally displaced people.
"We prayed to Allah to get us out of this hell. We were trapped in the battle area, but this morning we got a chance to flee," Fortun Mohamoud Iro, a mother of three, said, adding that she saw several bodies near a mosque.
Witnesses reported eight decomposing corpses lying near the Al-Hidaya mosque in northern Mogadishu and another was spotted near a stadium in the south of the capital.
Asad Mohamoud Moalim, a resident, told the AFP news agency that "no one dares go closer, let alone [conduct] burials".
Accurate casualty figures are difficult to establish because neither side has given an official toll and bodies remain uncollected because of the fighting.
Sudan Ali Ahmed, chairman of Elman Human Rights, said on Sunday that his group had tracked casualties through hospitals and morgues and found 81 fatalities.
He said that 119 people had also been wounded since fighting began on Saturday.
"The casualties ... were caused by Ethiopians using heavy artillery and tank shells in residential areas of the war-torn capital. We condemn this latest fighting," he said.
Health workers said that civilians appeared to have borne the brunt of the violence with scores of wounded patients being treated at hospitals across the city. 'Popular uprising'
The fighting on Saturday on Sunday was some of the heaviest in Mogadishu in recent months, according to locals.
Ethiopian troops helped the Somali military force the Islamic Courts Union movement out of the capital and much of the south of the country in late 2006. They have been deployed there ever since.
Mohammed Gure, chairman of the Somalia Concern Group, told Al Jazeera that the violence was a "popular uprising" against the Ethiopian forces.
"The fighters are not only from the Islamic courts, they are from the Somali nationalists, they come from the former Somali army, they also come from the children and the fathers of those who were displaced or had their homes and livelihoods destroyed by the Ethiopian tanks," he said.
"Somalia is a country under Ethiopian occupation and the Somali people are fighting to free their country."