Troops moved just south of Elephant Pass, the gateway to Jaffna and former army camp that the Tigers seized in 2000, and also took Oddosudan, a town on the road east to the Tigers' last major stronghold at Mullaittivu port, the military said.
"Troops captured the southern coast of Elephant Pass today," military spokesman Brigadier Udaya Nanayakkara said. The Tigers are dug into fortifications north of there, but are now stuck on a narrow strip of land with the army north and south of them.
Air force jets and attack helicopters bombed and rocketed LTTE positions near Mullaittivu and the A-34 road leading to it, the air force said.
The Liberation Tigers of Tamil Eelam (LTTE) had no immediate comment.
The military offensive has swiftly boxed the separatist Tigers into a wedge-shaped northeastern corner of the Indian Ocean island, after seizing their self-proclaimed capital of Kilinochchi on Friday.
Analysts say that was the most crushing defeat so far for the rebels, once viewed as one of the world's most resilient guerrilla groups but now increasingly losing ground to a determined Sri Lankan military.
Most analysts caution that if the conventional war ends, the LTTE is likely to revert to hit-and-run guerrilla tactics and bombings in the capital Colombo -- as it did on Friday, killing three airmen hours after Kilinochchi's capture was announced.
The Tigers are waging one of Asia's longest civil wars, with the aim of establishing a separate state for Sri Lanka's minority Tamils. The conflict has killed 70,000 since it began in 1983.
A year ago, President Mahinda Rajapaksa's government formally scrapped a poorly observed six-year truce brokered by Norway, accusing the rebels of using it to re-arm and vowing to destroy them unless they surrendered.
The LTTE says it is battling for the rights of ethnic Tamils mistreated by successive governments led by the Sinhalese majority since Sri Lanka won independence from Britain in 1948.
But rights groups say the LTTE, which is on U.S., EU and Indian terrorist lists, is increasingly using Tamils as human shields and forcibly conscripting them as fighters.
Aid agencies estimate at least 200,000 Tamils have been forced from their homes and are stuck between rebels who will not let them leave and a hard existence in military-guarded refugee camps where many are treated as possible LTTE suspects.