Bajaajlayaasha: Canshuur bixiyaal dhiigoodda la baneystay
The AU mission in Somalia (AMISOM) said it had captured the town of Bulomarer, some 160 kilometres (100 miles) southwest of the capital Mogadishu.
The town was the scene of an attempted raid by French commandos in January 2013 to free a secret agent being held hostage.
The bid failed and resulted in the death of two French soldiers and the hostage.
The new offensive is aimed at capturing key ports from the extremist group in an effort to cut off one of its main revenue sources.
"Operation Indian Ocean started late last night... The enemy is fleeing and the forces are making successful advances so far," said Abdukadir Mohamed Nur, the governor of southern Somalia's Lower Shabelle region.
Witnesses reported hearing booms from heavy shelling and seeing convoys of tanks and armoured vehicles southwest of Mogadishu.
AMISOM and Somali government troops were also seen on roads towards Barawe, the last major port held by the hardline gunmen on Somalia's Indian Ocean coast.
"The operations will not stop until the Al-Qaeda militants are eliminated," Nur added.
The Al-Qaeda-linked Shebab are still a powerful and dangerous force but have lost a string of towns to the 22,000-strong AU force.
Deployed in Somalia in 2007, AMISOM has made a series of advances in the past three years, but the launch Saturday of a new offensive targets the Shebab's lucrative charcoal trade.
- Hunger and drought -
In previous offensives by AU troops, Shebab fighters have fled in advance of the main column but later returned to stage guerrilla-style attacks.
Shebab fighters continue to launch attacks even in the heart of Mogadishu, including recent brazen commando raids on the presidential palace and parliament.
Ali Mohamed, who lives close to the Bulomarer, reported hearing "heavy shelling", while another resident, Hussein Mumin, described seeing military convoys including "several tanks."
Somalia's government in a statement claimed Shebab fighters were "fleeing in the face of advancing Somali and AMISOM forces," adding that Barawe was the "next objective".
The southern port of Barawe is now one of the few major settlements under their control, and is vital to Shebab finances, as the main hub of a multi-million dollar charcoal trade.
Charcoal, which is mainly exported to Gulf nations, generates at least $25 million (19 million euros) a year for the Shebab, according to UN estimates.
The Shebab are fighting to topple Somalia's internationally-backed government, and regularly launch attacks against state targets, as well as in neighbouring countries that contribute to the AU force.
The fighting comes as the United Nations and aid workers warn that large areas of Somalia are struggling with dire hunger and drought, three years after famine killed more than a quarter of a million people.
Somalia's government, selected in a UN-backed process in 2012, was widely hailed as offering the best chance in decades to repair the war-ravaged country.
But the return of extreme hunger, accusations of corruption and continued Shebab attacks in even the most heavily defended zones have cast a shadow over the government's record.
Bajaajlayaasha waxaa la dilaa iyaga oo bixinaya canshuurtii dowladda, waana cadaalad daridda ka jirta Muqdisho sida uu Farmaajo xilka ula wareegay oo ku kalliftay dhalinyarro badan in ay gabi ahaamba dalka isaga tagaan.