There was no immediate comment from the rebels.
Vakarai, a town around 240 km northeast of Colombo, belonged to the rebels under the terms of a now seemingly defunct 2002 ceasefire.
The military have said they aim to evict them from the east altogether. "They are withdrawing," Samarasinghe said.
"We are checking the civilians who are fleeing in case the Tigers have infiltrated them and try to come and attack government areas."
He said some clusters of Tigers were trying to head north towards the neighbouring district of Trincomalee.
The military said they had recovered one rebel body so far, and that nine other Tiger fighters were killed in a separate clash further south near Batticaloa.
|"These people are obviously weak and afraid and we are looking for assurances their protection will be assured."|
Orla Clinton, UN
Some on foot, others on tractors, the refugees left the rebel-held town of Vakarai on the island's northeast coast at dawn and were heading south towards government-held territory a few miles away, some carrying white flags.
"Some of the people are making their way down the coast, others are coming through jungle in vehicles," Selvaraj Jeyaraj, project coordinator for the Italian Red Cross, told Reuters by telephone from the eastern district of Batticaloa.
"We are talking about 12,000 internally displaced, more or less," he added.
The UN confirmed the exodus.
About 20,000 to 25,000 refugees had already fled Vakarai in recent weeks.
The area around Vakarai is one of several pockets of territory the Tigers control in the island's east and is their last remaining direct access to the sea in the east, an important supply line.
The Tigers' patches of eastern territory are cut off from their main northern stronghold by military-held areas.
The army has accused the Tigers of using displaced ethnic Tamils as human shields and planting thousands of landmines to prevent them leaving.
But there has been no independent confirmation any mines had been laid.
"Thousands of people are streaming out," Orla Clinton, a spokeswoman for the UN, said.
"Because we haven't had access [to Vakarai] since November 29, food has been very short - these people are obviously weak and afraid and we are looking for assurances their protection will be assured."
Aid agencies, embassies and journalists have been requesting access to rebel-held areas for several weeks but the government has refused.