The latest clashes follow Tuesday's deadly attack on a hotel in Mogadishu in which 38 people, including eight members of parliament, died.
The US condemned the attack in which gunmen wearing military uniforms stormed the Muna Hotel, that hosts government officials, before a suicide bomber blew himself up inside.
Al-Shabab said that its special forces carried out the attack.
The assault came after the group declared a "decisive" war against African Union peacekeepers from Uganda and Burundi deployed in the city.
But Abdirahman Yariisow, Somalia's information minister, told Al Jazeera that al-Shabab will not defeat his government.
"We are winning this war ... yesterday's suicide attack was some kind of revenge for the losses [we inflicted on al-Shabab]," he said.
"Al-Shabab are terrorists, they are working for al-Qaeda and they have no regard for civilians."
Sheikh Ali Mohamoud Rage, al-Shabab's spokesman, said fighters were starting a new war against "invaders" - an apparent reference to the 6,000 African Union peacekeepers deployed in the country to support government forces.
Hundreds of new troops, mostly Ugandan, have arrived in recent days to help the government in its battle against al-Shabab.
The AU force has so far been able to do little more than guard the airport and port and shield Sharif Ahmed, the president.
Somalia's members of parliament, the apparent targets of Tuesday's attack, do not benefit from this protection.
Uganda said last month that it was willing to send 1,200 troops to Somalia to supplement the existing 6,000-strong Amisom mission.
Al-Shabab, which has been fighting Somalia's government since 2007, recently claimed responsibility for a twin bombing attack in Uganda,which killed more than 70 people who were watching the football World Cup.
The group has said that it will continue to carry out attacks in Uganda and Burundi as long as those countries provide troops for the AU peacekeeping force in Somalia.
Somalia has not had an effective central government for nearly 20 years and al-Shabab controls significant portions of the country's south.
More than 21,000 Somalis have been killed in fighting since the start of the uprising, 1.5 million have been uprooted from their homes and nearly half a million are sheltering in other countries in the region.