"Any video or site just pops up... they're very excited about the speed," Somalia Wireless's Liban Egal says.
Until now access to the internet has been via dial-up or satellite links.
Earlier this year, 3G mobile phone services were cut off because of a threat from Islamist militants.
The al-Qaeda-linked al-Shabab group issued a directive in January ordering all internet services to be stopped, saying those who did not comply would be seen as "working with the enemy" and dealt with according to Islamic law.Al-Shabab was driven out of the capital, Mogadishu, in August 2011, but still controls many smaller towns and rural areas in the south and centre of the country where they have imposed a strict version of Sharia.
Following their threat, 3G networks nationwide were turned off but the project to launch fibre optic cable services continued in the capital, the BBC's Moalimu Mohammed reports from Mogadishu.
'Day and night'He says the fibre optic connections, which have been rolled out over the last week by several internet providers, are only available in Mogadishu.
People have been flocking to hotels and internet cafes to try out the fast service - some seeing video platforms like YouTube and social networking sites for the first time, our correspondent says.
Mr Egal said the difference in speed was like the difference between "day and night".
For those residents who have recently returned from the diaspora the development was a relief, he said.
It was "almost a culture shock" for those who have never left Somalia, he added.
He said the move would be a huge boost for the country, which is recovering from more than two decades of civil war.
"Every time a fibre optic cable is connected to a country they see their GDP [gross domestic product] going up because their communication costs go down," Mr Egal said.
"All life will be affected - businesses, the government, universities - they all will see the benefits."
Our reporter says the current security situation will limit the rollout of fibre optic services to the rest of the country.
Since 1991 Somalia has seen clan-based warlords, rival politicians and Islamist militants battle for control - a situation that has allowed lawlessness to flourish.
An African Union force has been helping the UN-backed government fight al-Shabab, which wants to create an Islamic state in Somalia.