Showing AFP aerial video from a military spy plane over the tiny area where the Tigers are staging a last stand, Rajapakse said 35,000 non-combatants had crossed the lines into government-held territory within a five-hour period.
The military said the surge of civilians had continued 10 hours after the initial exodus of men, women and children.
"The footage clearly shows that the people are defying the LTTE (Liberation Tigers of Tamil Eelam) and escaping. They are running to safety," the president said.
"What we are doing is not a military operation, but the world's biggest hostage rescue."
The government has accused the LTTE of using trapped civilians as a human shield, and the president suggested that their escape removed a final obstacle to an all-out military assault.
"The process of the complete defeat of the LTTE has just begun," he told AFP.
"It is now all over for the Tigers.
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The live streaming video from the spy plane showed hordes of people running towards m