The Sri Lankan army has for months been advancing towards Kilinochchi, which has been in the hands of rebels for the last decade.
Both sides have recently claimed to have inflicted heavy casualties on each other in the north of the island.
But there have been no independent reports from the front line and it is impossible to verify either account of casualties.
"We should pay the gratitude of the whole nation to those heroic soldiers who achieved that victory," President Rajapaksa said in a nationally televised address.
He urged the rebel fighters to lay down their arms.
Sri Lankan troops fighting in the north of the country
The pro-Tamil website TamilNet reported the occupation of Kilinochchi, saying troops had entered a "virtual ghost town" as the civilian population and rebels shifted further north-east.
It said most of the buildings in the town had been destroyed by continuous military strikes and added that rebel casualties had been kept low despite the fighting.
In Colombo, a suspected suicide bomber attacked the headquarters of the Sri Lankan air force, killing at least two personnel and wounding around 30 people, including nine airmen, officials say.
The scene of the bomb blast outside Sri Lanka's air force headquarters
Earlier on Friday, government officials said troops had entered Kilinochchi from three directions and predicted the town would fall within a few hours.
Military spokesman Brigadier Udaya Nanayakkara said troops had to overcome "enemy pockets" of rebel fighters.
On Thursday, the military said it had seized the strategically important junction of Paranthan, a crossroads north of Kilinochchi, in a bitter fight that lasted for hours.
It said the success at Paranthan had effectively cut the main supply line to several Tiger strongholds in the north of Sri Lanka.
Although the loss of Kilinochchi will be a blow to the Tigers, the head of its political wing, B Nadesan, told the BBC recently they would be able to continue fighting even if they lost the town.
The rebels would remain in possession of some territory to the east of the town down to Mullaitivu on the coast, although that too is under threat from government forces.
Correspondents say that while the government seems able to maintain the upper hand, heavy battles are likely still to lie ahead and there is concern about the fate of the large number of civilians in the Tiger-controlled north.
The rebels deny using them as human shields and reject allegations they are forcing people into their ranks to fight.
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