"I can confirm at this time that we managed to outdo the enemy," senior police official Ali Abdi told The Anadolu Agency.
"We killed three Al-Shabaab terrorists. Four government soldiers and three Ethiopian AU soldiers were also killed," he added.
According to police, a bomb exploded outside the palace of the regional president. Minutes later, armed men stormed the compound and started firing indiscriminately.
Three civilians caught up in the gunfight were injured.
South West State President Sharif Hassan Sheikh Aden, who was reportedly present during the attack, is said to have been unharmed.
The Al-Qaeda-linked Al-Shabaab group later claimed responsibility for the attack, asserting that seven Ethiopian African Union peacekeepers had been killed.
Thursday's bombings come one day after twin blasts rocked a Mogadishu hotel, injuring at least four people.
Fractious Somalia has appeared to inch closer to stability since government troops and African Union forces – deployed in the country since 2007 – drove the Al-Shabaab militant group from most of its strongholds.
Nevertheless, the group has continued to stage attacks on government forces and African Union peacekeepers.