The military said soldiers pushed deep into Mullaittivu, the last remaining rebel stronghold, and seized a training camp. It did not provide casualty details.
It was not possible to independently verify the military's claims, as journalists are banned from the war zone.
"The military say they are taking territory at about a kilometre a day," Al Jazeera's Tony Birtley reported from Sri Lanka.
"But there are no heavy weapons that have been captured by the military. We know that for some time the Tigers have been stockpiling weapons - they have got big weapons, they have got artillery - and this suggests the Tigers are still a capable fighting force," he said.
"The military is taking territory but there still is a tremendous capability, we think, from the Tigers."
As fighting intensifies, aid groups and diplomats have expressed fears for the safety of hundreds of thousands of civilians reportedly trapped in Tiger-held territory.
The rebel-affiliated TamilNet website said that five civilians were killed on Friday and 83 wounded when the army fired artillery shells into a government-declared "safe zone" for displaced families.
A doctor in the area confirmed on Saturday that five civilians were killed in the shelling.
The military denied firing into the civilian settlements and launching attacks on the "safe zone", accusing the Tigers of carrying out the assaults themselves to keep civilians out of the area.
Human rights organisations have accused the rebels of using the civilians as human shields to block the government offensive.