The soldiers will eventually make up part of a 8,000-strong AU peacekeeping force for the Horn of Africa nation.
Captain Paddy Ankunda, the spokesman of the AU contingent, told AFP news agency: "I believe every boot on the ground will change the situation and we hope other countries contributing solders will take the same path as Burundi and will deploy their forces soon."
Nigeria is also due to send troops to Somalia in the next two or three months.
Meanwhile, overnight battles in the north of Mogadishu left at least four people dead.
Two reportedly died when a mortar shell landed on a house, while two more died in fighting, locals told Reuters news agency.
About 6,000 civilians have been killed and hundreds of thousands forced to abandon their homes in the past year as Somalia's week interim government has struggled to hold on to power.
Numerous bids to restore stability to the Horn of Africa nation since the 1991 ousting of Mohamed Siad Barre, the former president, have failed because of clan warfare and unrest.