The port city has been a stronghold of the al-Qaeda-aligned group al-Shabab.
Al-Shabab spokesmen told news agencies that fierce fighting was now under way.
The Kenyan troops are part of the AU’s Amisom force, which is trying to wrest control of the country for the newly elected UN-backed president.
Kenyan military spokesman Col Cyrus Oguna confirmed to the BBC that parts of Kismayo had been captured and the rest was expected to fall soon.
Col Oguna said the joint operation of Kenyan Defence Forces and Somali government troops had begun at 02:00 local time (23:00 GMT Thursday) and was “basically amphibious”.
Col Oguna said: “We cannot give casualty figures at the moment, the damage has not been assessed, but I can tell you our forces are already in Kismayo.”
He told the BBC: “There are some parts that still will be under the control of al-Shabab because we only got there a couple of hours ago, and Kismayo is a big city.”
Amisom released a statement saying it had “successfully inserted” troops “to liberate the people of Kismayo to enable them to lead their lives in peace, stability and security”.
It urged “all fighters remaining in Kismayo to lay down their arms”.
Al-Shabab spokesmen said fierce clashes were taking place.
Al-Shabab commander in Kismayo, Sheik Mohamed Abu-Fatuma, told Agence France-Presse news agency: “The enemy using military boats have deployed hundreds of soldiers in the coast late last night and the mujahideen fighters are engaging heavy fighting now with them. God willing they will be defeated.”
Mohamed Mohamed, of the BBC’s Somalia Service, has spoken to residents of Kismayo, who said four military ships had approached from the north, landed and captured territory around the university.
Checkpoints were set up, with one eyewitness reporting that the AU forces appeared to include American or European troops. There have been numerous reports of US special forces operating against the Islamist militants in Somalia.
However, the US Africa command said on Friday it was “not participating in Kenya’s military activities in the region”.
The Kismayo residents told the BBC they had seen no Kenyan or Somali government forces inside the city proper.
Other residents told Reuters news agency they could hear fighting outside the city.
One, Ismail Suglow, told the agency: “Al-Shabab have gone towards the beach. Many residents have taken their guns. The ships poured many AU troops on the beach.”
There are also reports that helicopters are attacking the town.
The al-Shabab-controlled radio station is still said to be on air.
Al-Shabab also used a Twitter account to say its forces were still in control of the city.
Earlier this week, Kenyan military jets had bombed the airport in Kismayo, destroying an armoury and warehouse used by Islamist militants.
Some 10,000 people have fled Kismayo in the past week, the United Nations refugee agency has estimated.
On Thursday, leaflets were dropped urging remaining civilians to evacuate.
BBC World Service News Africa editor, Martin Plaut, says al-Shabab has tried to halt the exodus and has urged civilians to join their fight.
He says the advance by Kenyan forces, Somali government troops and the pro-government Ras Kambone militia has been delayed as all roads into the city had to be swept inch by inch for mines. This, our correspondent says, explains the attack from the sea.
Kenya began its intervention in Somalia nearly a year ago after a spate of cross-border attacks blamed on al-Shabab.
Al-Shabab has been forced out of the capital, Mogadishu, and several other towns over the past year but still controls much of the countryside in south and central Somalia.
Since the overthrow of President Siad Barre in 1991, Somalia has seen clan-based warlords, Islamist militants and its neighbours all battling for control.