There was no immediate response from the rebels, who have rejected previous calls to surrender. The pro-rebel TamilNet website said several hundred civilians were feared killed and injured after troops advanced into the zone.
Each side accuses the other of killing civilians in the long running civil conflict.
Foreign reporters are not allowed into the combat zone, making it impossible to independently verify the claims.
A doctor speaking to the BBC from the conflict zone said that one hospital had been hit by shelling and gunfire early on Monday.
Dr T Satyamurthy, director of health services in Kilinochchi, said there were several hundred injured and dozens of dead there and at another, nearby hospital.
He also said his staff had told him that they had seen "many dead bodies" by the edge of a road in the area.
The BBC's Charles Haviland in Colombo has seen the video, shot by an unmanned military vehicle, of civilians fleeing the combat zone. The film was apparently taken early on Monday morning.
BBC News, Colombo
The government and the military are feeling increasingly bullish.
They felt able to call in a large number of journalists and show aerial footage shot from an unmanned plane.
The pictures are remarkable. Hundreds or thousands of people, small specks in the camera frame, are seen hurrying in a direction away from the sea, across land, lagoon and swamp.
Other pictures show what looks like tens of thousands milling around, waiting to be processed at army checkpoints.
Confirming estimates of at least 25,000 people fleeing, our correspondent described as "haunting" the sight of thousands of people filing as quickly as they could away from the ocean side of the strip of land where fighting has been going on.
A far greater number were visible milling around outside the zone waiting to be taken to government-run camps, our correspondent says.
Some estimates say as many as 35,000 people fled the fighting on Monday.
President Mahinda Rajapakse said: "The footage clearly shows that the people are defying the LTTE [Liberation Tigers of Tamil Eelam] and escaping. They are running to safety.
"The only thing [LTTE leader Velupillai] Prabhakaran can now do is to surrender."
Earlier in the day, military spokesman Brigadier Udaya Nanayakkara said the army had managed to smash through an earth fortification which had been blocking its advance into the tiny coastal strip held by the rebels.
He said those who had just escaped would go, like others, to special camps around the town of Vavuniya about 100km (60 miles) from the fighting.
The Tigers are restricted to a 20 sq km (12.4 sq miles) coastal patch that the government has designated a "safe zone" for civilians.
Gordon Weis, the UN spokesman in Sri Lanka, said it was not known how many civilians remained there but that the UN had been working off a figure of some 150,000 to 200,000 people in recent months.
Hospitals in the war zone are lacking staff and medical supplies
"It's still a significant number of people locked inside," he told the BBC.
Our correspondent says life for the Tamil civilians in the zone is a nightmare.
There has been shelling for months, while the UN says the Tigers are preventing people from escaping, despite rebel denials.
A military spokesman said 17 people were killed on Monday when the rebels launched three suicide attacks on those fleeing, though it was not possible to verify his account.
The government is not giving the International Committee of the Red Cross (ICRC) access to the landward side of the zone.
So it can only evacuate people by sea, with two or three ships per week each carrying 400 or 500 of the sickest, oldest and most badly wounded people.
ICRC country chief Paul Castella told the BBC that government doctors working in the area were "really worn out".
"These people are working for months now without any break, they work day and night. And medical supplies are lacking," he added.
The Red Cross has evacuated more than 10,000 civilians since early February, but Mr Castella could not confirm or deny figures that others are citing for the number of people recently killed, or managing to escape by land.
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