The government warned the rebels they had 24 hours to surrender or face a final assault to end a crumbling 25-year insurgency that sought to create a separate homeland for ethnic Tamils on this South Asian island.
President Mahinda Rajapaksa went on Sri Lanka's national television to say that soldiers helped more than 35,000 civilians leave the battle zone in what he called the "largest-ever hostage rescue mission in history."
The Red Cross said its workers had tended to 4,000 people who crossed the front lines Monday. Spokeswoman Sarasi Wijeratne said the organization was not in a position to "confirm or deny" the large number being quoted.
A pro-rebel website, meanwhile, said hundreds of civilians might have been killed in the "total chaos" that prevailed when soldiers entered the zone.
It was not possible to verify any of the reports because journalists are restricted in the war zone.
U.N. Secretary-General Ban Ki-moon welcomed the civilians' escape but remained concerned about thousands still trapped and "the potential for large-scale casualties," U.N. spokeswoman Marie Okabe said in New York.
Ban also said aid workers must be allowed into the area to help civilians.
U.S. State Department spokesman Robert Wood called the humanitarian situation "dire" and asked both sides to "cease this violent activity."
The U.N. had estimated 100,000 civilians were trapped in the zone where the rebels have been pinned down, an area that measures less than 8 square miles.
U.N. officials say 4,500 noncombatants have been killed over last three months amid fierce fighting during a government offensive that has driven the rebels from their strongholds.
The military spokesman, Brig. Udaya Nanayakkara, said soldiers broke through part of a 7-mile-long earthen barrier erected by the Tamil Tigers to seal off the small area that had been designated a "no-fire zone" and where the rebels have dug in for a final stand.
That allowed civilians to stream out toward government lines.
Video on state television Rupavahini showed dozens of men and women wading across a lagoon in neck-high water. Some people were wounded and dripping blood. Others carried children or wounded relatives on their shoulders.
A girl wept over the body of a relative who state television said was killed in a rebel suicide bombing aimed at preventing civilians from leaving.
Few possessions were carried out. Some people wore rucksacks on their backs, others held sacks on their shoulders or bulkier bags on their heads.
"We couldn't move from there. In all directions the LTTE (Liberation Tigers of Tamil Eelam) cadres were surrounding and searching everywhere. If they found someone planning to escape they will immediately shoot them down," a man named Murugan told state television.
The air force showed reporters video taken by a surveillance plane in which dozens of civilians ran toward the army's front line, most of them carrying bags on their heads.
Footage shot by Associated Press Television News showed men, women and children resting on a beach at Puttumattalan after fleeing the war zone.
The military said the vast majority of those who fled Monday -- more than 25,000 -- headed to an army-controlled area where they were being screened. The military spokesman said 2,165 people also escaped in 103 boats and were picked up by the navy at sea.
The exodus came just days after the military announced a unilateral, two-day cease-fire to encourage civilians to flee. Only a few hundred left initially, prompting the government to again accuse the rebels of using civilians as human shields, a charge also made by aid groups.
Tamil rebels have previously denied forcing civilians to stay, but it was not possible to contact the rebels for comment Monday.
The U.N. and others have called for a negotiated truce to allow civilians to leave. But the government has rejected such calls, saying it is on the verge of crushing the rebels after an insurgency that caused some 70,000 deaths since it began in 1983, fed by Tamil anger over decades of marginalization by the island's ethnic Sinhalese majority.
The government said Monday that rebel leader Velupillai Prabhakaran and his fighters had 24 hours to surrender before a final assault -- one of many such recent claims that troops would soon end the conflict.
Defense spokesman Keheliya Rambukwella said the government's preferred option was to catch Prabhakaran alive and the president's ultimatum was a final opportunity for the leader to end the conflict.
The rebels have refused previous calls for their surrender.
Rambukwella said the rebel leader's capture or death was "inevitable" because he would soon lose his civilian cover. "He doesn't have that option now," the spokesman said.
Military analyst Susantha Seneviratne, a retired colonel said the call for surrender was a way to probe the morale of rank-and-file rebels.
The army has "pressured them and it has been constant, and this is a method to test how they react," he said.
"As far as the LTTE (leadership) is concerned, they will not surrender," he predicted.