Puntland usually enforces a secular penal code, even though the region's charter says it is based on the Quran.
Puntland, which declared itself an autonomous state within Somalia in 1998, has generally been spared the violence that has wracked much of the country.
"Our decision that to use Islamic law will be different from the type of militant Islam in Mogadishu," where the Islamic Courts rule, Museh said.
"We do not want to politicise the Islamic religion."
Puntland's promise to resist the Islamists' expansionist ambitions comes amid reports of brief but heavy outbreaks of fighting around Baidoa, the headquarters of the country's nominal government.
Witnesses in a town about 60km from Baidoa said on Tuesday that Ethiopian troops had clashed with forces loyal to the Islamists, and that three Ethiopian soldiers were killed overnight.
These reports have not been confirmed by the Ethiopian government.
Ethiopia, a mainly-Christian country, says that it provides training and logistical support to the interim government in Baidoa.
The Islamists accuse Ethiopia of deploying thousands of troops inside Somalia to protect the interim government from attack.
A turbulent history
Somalia has not had an effective government since 1991, when warlords overthrew a dictator and then turned on each other.
The current administration was formed with the help of the UN two years ago, but it has failed to assert any real control outside the southern town of Baidoa, where it is based.
The Islamic Courts have steadily gained ground since taking over the capital, Mogadishu, in June and now control much of southern Somalia.