The army is seeking to disrupt Tamil Tiger supplies
Thousands of Tamil civilians are fleeing the town of Vakarai, which has been the focus of heavy fighting for several weeks.
Aid agencies say they are worried about the fate of the civilians, many of whom were taking shelter in a hospital in Vakarai affected by the fighting.
The fall of Vakarai would badly disrupt Tamil Tiger supplies to the north.
There has been no comment from the Tigers on the military's claim.
Fled
"The troops are in control of the town of Vakarai," military spokesman Prasad Samarasinghe told the BBC's Sinhala service. "Thousands of civilians are coming across to government-controlled areas."
The defence ministry said at least 45 members of the security forces and 331 rebels had been killed in the battle for Vakarai, which began in October.
The BBC's Roland Buerk in Colombo says Tamil Tiger fighters fled into the jungle and by boat as government forces moved in.
The pro-Tamil Tiger website Tamilnet says that earlier the military stepped up shelling of the hospital in Vakarai, "forcing 15,000 civilians to flee... the hospital area".
The military has accused the Tigers of using civilians in the hospital as human shields and using the hospital as "an artillery launching pad".
Before Friday's developments, more than 30,000 civilians had already left Vakarai in recent weeks to escape the conflict.
The government says thousands more moved into areas under its control on Friday.
The Tigers want an independent homeland for Tamils
The United Nations Refugee agency, UNHCR, says more may remain in Vakarai and it is extremely concerned for their safety.
The UNHCR is calling on all sides to respect human rights.
Our correspondent says Vakarai was a key stronghold of the Tigers, used as a transit point between Sri Lanka's east and the north.
The Tigers had their own police force and civil administration in the area.
Homeland
In recent months the government has been stepping up military activity in the east, aiming to seize pockets of territory held by the Tigers.
About 65,000 people have been killed in Sri Lanka's civil war.
The Tigers are fighting for a separate homeland for the country's 3.1m minority Tamils, who say they have suffered decades of discrimination by the majority Sinhalese.
Although both sides say they are adhering to a Norwegian-brokered 2002 cease-fire, violence has escalated since late 2005, with more than 3,600 people killed last year alone.
Sri Lankan troops liberated a key town in Eastern Sri Lanka as they continued their offensive to evict the Liberation Tigers of Tamil Eelam (LTTE, also called the Tamil Tigers) from the Eastern Province of the country. 

The town of Vakarai, which has been under the control of the LTTE since 1996, was a key stronghold for the LTTE and one of their last major bases in the East. Sri Lankan troops met little resistance as they moved into the town at around noon local time, as the Tamil Tigers were believed to have fled further North.

A Sri Lankan defense ministry statement on its website said the LTTE had fled the town area "unable to withstand the intrusive thrust of the armed forces". A statement from the Media Center for National Security said "triumphant troops are presently consolidating their positions and clearing the terrain to resettle displace civilians."

The Tigers have so far failed to comment on the fighting. 

Leading up to the capture of the town, around 12,000 civilians, whom the government claimed the LTTE were using as Human Shields, were reported to have fled to the safety of government controlled areas. This added to the 20,000 who had already fled to government controlled areas since heavy fighting began in December 2006.

Basic necessities for them were provided by the security forces. The government stated that it would make all efforts to resettle the civilians as soon as normalcy is restored in the town.

The fall of Vakarai is expected to badly disrupt Tamil Tiger activities in the East, an also seriously dent their supplies to the North.