"The government advanced to a 3km earth barrier on the edge of the no-fire zone and destroyed it this morning.
"At least 13,000 Tamil civilians who were held as hostage by the LTTE [Liberation Tigers of Tamil Eelam] in the no-fire zone have escaped into the government-held safety zone just beyond the lagoon from the small stretch of land that the LTTE holds.
"We are told there are more people coming ... I suppose the government will be in a position to declare military victory very soon."
The earth wall had blocked the widest land link to the coastal strip where Sri Lanka's military has surrounded the remaining separatists.
The rescue operation was ongoing and more civilians were being brought out of the area, the defence ministry said.
Velupillai Prabhakaran, the LTTE leader, has been given until noon on Tuesday (17:30 GMT) to surrender or face a "military course of action."
Mahinda Rajapaksa, Sri Lanka's president, said time had run out for Prabhakaran, who has not been seen at the LTTE's public functions for nearly 18 months.
"The only thing Prabhakaran can now do is to surrender," the president said.
"I don't want him to take cyanide and commit suicide. He has to face charges for his actions."
Sri Lankan media have been speculating that he may already have fled the country by boat.
Al Jazeera's David Chater, reporting from Colombo, said defence ministry officials told him there were about 300 Tamil Tigers left in the "no-fire zone", armed with heavy weapons.
"They also say Tamil Tigers have infiltrated the civilian population and that is why we saw the suicide bombings today.
"As the civilians flee, the suicide bombers are inside the groups and detonate their devices as they're coming out," Chater reported.
There was no immediate comment from the LTTE about the reports and it is impossible to independently confirm the military's reports as journalists are banned from the conflict zone.
Before Monday's reported military operation, aid groups said more than 100,000 civilians were trapped between the army and Tamil Tiger fighters in the "no-fire zone".
The United Nations has repeatedly accused the military of shelling the area, and the Tamil Tigers of preventing civilians leaving, effectively using them as human shields.
Both sides reject the claims.
The United Nations says at least 4,500 civilians have been killed in the last three months of fighting, and another 12,000 injured.