Mr Salat who is his country's only elected president in a decade arrived with his newly-appointed prime minister, Ali Khalif Galayadh, and members of his transitional government.
He was appointed by a number of clan elders and other senior figures at a special conference in Djibouti in August with the help of East African leaders, but not all the warring Somali factions accept him as a legitimate choice.
A convoy of about 100 vehicles carrying Mr Salat and his entourage drove through the streets of north Mogadishu and included 1,000 heavily-armed security guards.
Hero's welcome
The president was welcomed at a military airstrip just outside the capital and was treated to a national anthem and a red carpet.
Children climbed trees and lined the streets as women dressed in traditional clothes sang his welcome.
Others lined the streets waving flags and flowers to welcome the new president.
Mr Salat's motorcade raced through the streets of north Mogadishu to hotel Ramadan which will be his temporary home.
One enthusiastic teenager, 14-year-old, Abdallah Issaq summed up what the war-torn country needed.
"We need a president and security. There is no education. We need schools," he said as he waved a flag to welcome the president.
Estimates say up to a million people have been killed in a civil war among the divided clans of Somalia.
Future challenges
After several failed peace conferences Somalia's 245 member parliament elected Mr Salat at a specially convened conference in Djibouti.
Prime Minister Ali Khalif Galayadh has promised to announce his cabinet in a few days.
Speaking to journalists before arriving with the president in Mogadishu, he admitted that he faced a daunting challenge.
"I think we have a handle on the issue of security. There may be some freelance elements, or one or two hotheads, but as far as we can tell there is no organised resistance," he said.
Most of the clans from the north of the country say they will not recognise the government.
The BBC's East Africa correspondent, Cathy Jenkins, says after decades of rule, most warlords don't want to give up their power.