The central government has declared a state of emergency in the region, and has appealed for international aid.
Thousands of livestock have also died and hundreds of homes have been destroyed by the cyclone, known as 03A.
The cyclone has subsided, but officials say heavy rains are still expected.
Puntland is a semi-autonomous region within Somalia and a large part of its population relies on livestock herding and fishing.
The tropical cyclone swept through the Eyl, Beyla, Dangorayo and Hafun districts along the east coast and across to Alula on the tip of the Horn of Africa.
"So far we have confirmed the storm killed 140 people. We are afraid the death toll may reach 300 because many people are still missing," Puntland's Interior Minister Abdullahi Ahmed told Reuters news agency.
A government statement said Puntland's security forces had failed to transport 60 tonnes of aid to affected communities because heavy flooding had made many dirt roads to the worst-affected areas impassable.
"The storm has destroyed entire villages, homes, buildings and boats," the statement added.
Makeshift shelters had been built to accommodate displaced people, it said.
The government has appealed to international aid agencies to help with air-drops of relief supplies, including clean water, non-perishable food, tents, blankets and medicines.
Ten doctors and two ambulances were attending to people in need of medical treatment, the statement said.
On Tuesday, Somalia's government pledged $1m (??628,000) to help communities devastated by the cyclone.
The central government "shared the pain" of communities, Prime Minister Abdi Farah Shirdon said.
A cyclone that hit Somalia's north-eastern Puntland region at the weekend is known to have killed 140 people, and the number could rise to 300, a government minister has said.