The announcement came after the government said earlier in the day that the army had entered the town and hoped to secure it "in hours".
Heavy rebel resistance was reported as small groups of special forces entered Mullaittivu backed by 50,000 soldiers and helicopter gunships.
End of war?
The military has made dramatic gains in recent months, pushing the Tamil Tigers, who are fighting for an independent homeland in the north and east of the island, back to a small area in the east of the country.
The Tamils are an ethnic group found mainly in Southern India and Sri Lanka.
Population: Tamils comprise about 10 per cent of Sri Lanka's 21 million people
Inequality: The Tamils say they have been subject to persecution and discrimination by the country's Sinhalese majority, including restricted access to farming land, fishing areas and food
Demands: The Tamil Tigers want a self-governed state for Tamils
Earlier this month, the Sri Lankan military captured the town of Kilinochchi, the Tamil Tigers' de facto capital in the north of the island.
The Tamil Tigers have not commented on the government's claims.
In the capital Colombo, some Sri Lankans responded to the news that the 25-year civil war could be over by honking their car horns and lighting firecrackers.
But Tony Birtley, Al Jazeera's correspondent reporting from Colombo, said the fall of Mullaittivu might not halt the country's long-running conflict.
"The LTTE are still strong, they still have a lot of backing and maybe we haven't seen the end of the Tamil Tigers yet. The country is practically down on its knees, not only because of the cost of executing this war, but because of the loss of revenue from tourism which was a major earner for Sri Lanka.
"But the issues that have caused this war 25 years ago have not been resolved and some of the critics of the government say there is a need for a political solution. That seems to be put on the side and at the moment, it seems to be military solution."
Birtley said earlier that the rebels could flee to the jungles and carry on fighting.
The rebels took control of Mullaittivu in 1996 when they overran a military camp there, killing nearly 1,000 soldiers.
On Sunday, the military said that the retreating rebels had flooded two villages after destroying a reservoir in an attempt to stall advancing government troops.
"As the conflict lines get closer these people are becoming more and more exposed to intense fighting and are increasingly caught in the crossfire"
James Elder, spokesman for the UN in Sri Lanka
Tamil Tiger fighters used explosives to destroy the walls of Kalmadukulam reservoir on Saturday, as government troops advanced on Visuamdu, in Mullaittivu district, a statement said.
Details of the welfare of villagers and the damage caused by flooding from the dam were not immediately available.
Soldiers also clashed with fighters in Chundikulam village in the same district and hours later recovered the bodies of two Tamil Tiger fighters, according to the statement.
As fighting has intensified, aid groups and diplomats have expressed fears for the safety of hundreds of thousands of civilians reportedly trapped in Tiger-held territory around Mullaittivu.
"As the conflict lines get closer these people are becoming more and more exposed to intense fighting and are increasingly caught in the crossfire," James Elder, a spokesman for the UN in Sri Lanka, told Al Jazeera.
"We are calling on the LTTE [Liberation Tigers of Tamil Eelam] to meet their international responsibilities and guarantee that the civilian population can move freely and get away from this conflict zone," he said.
The rebel-affiliated TamilNet website said that five civilians were killed on Friday and 83 wounded when the army fired artillery shells into a government-declared "safe zone" for displaced families.
A doctor in the area confirmed on Saturday that five civilians were killed in shelling.
The military denied firing into the civilian settlements and launching attacks on the "safe zone", accusing the Tigers of carrying out the assaults themselves to keep civilians out of the area.
Human rights organisations have accused the rebels of using the civilians as human shields to block the government offensive.