The LTTE statement read: "For over three decades, our leader was the heart and soul and the symbol of hope, pride and determination for the whole nation of people of Tamil Eelam.
"Since the failure of the peace process and the escalation of the war forced upon the Tamil people, the LTTE was faced [sic] to confront the Sri Lankan military that was supported by the world powers.
"This deliberate bias and position taken by the international community severely weakened the military position of the LTTE.
"Our leader confronted this threat without any hesitation. He would not waver in his desire to be with his people and fight for his people til the end.
"His final request was for the struggle to continue until we achieved the freedom for his people.
"His legend and the historical status as the Greatest Tamil Leader ever are indestructible."
The Sri Lankan authorities have not published a post mortem examination report or officially confirmed how or when Prabhakaran died.
A government spokesman said on Tuesday that the body would be given to an undertaker, but General Sarath Fonseka, Sri Lanka's army chief, told the privately-run Sunday Rivira newspaper that it had been cremated and the ashes thrown in the sea.
Fonseka said Pottu Amman, the LTTE intelligence chief, and Prabhakaran's wife, Mathiwadini, were among the dead, but have yet to be officially identified.
The government released footage of Prabhakaran's body for the first time on Tuesday after Tamilnet, a pro-LTTE website, denied the government's announcement that he had been killed.
Sri Lanka said on Sunday it would not allow aid workers complete access to civilians who remain held in camps after the conflict, saying LTTE remnants still remained among the refugees.
Ban Ki-moon, the UN secretary-general, on a visit to a camp housing 200,000 Tamils, had called for his staff to be given "unhindered access" to those displaced in the decades-long war that ended last week.
"His legend and the historical status as the Greatest Tamil Leader ever are indestructible"
Selvarasa Pathmanathan, LTTE international relations chief
The government said that "as conditions improved, especially with regard to security, there would be no objections to such assistance".
Ban, who toured the Menik Farm facility on Saturday, described the conditions as overcrowded and the detained civilians as "badly in need of food, water and sanitation".
The government has described the camps as "welfare villages" and says it wants to resettle all displaced civilians as soon as possible.
But Tamil activists say they are "concentration camps" with inmates penned in behind barbed wire.
During his visit, Ban urged Rajapakse to probe alleged human rights violations committed during the defeat of the Tamil separatists, a joint statement on Sunday said.
The UN estimates that up to 7,000 civilians have been killed in fighting since the beginning of the year.