"Troops are now poised at launching a decisive blow at the remaining LTTE strong points at Mulliyan, Chempiyanpattu, Chundikulam and Kaddaikadu."
Tamil Tiger fighters have reportedly retreated to the town of Mullaittivu following months of government offensives in the north.
Soldiers advancing on the town captured a 2.5km runway on Saturday that the LTTE had used to launch bombing raids on military and economic targets, the military said in a statement.
"The end-game of LTTE's protracted separatist cause is reaching its final stages, as the advancing security forces overran the most fortified LTTE northern garrison," the ministry said.
Military analysts say that Tiger forces have been steadily withdrawing, moving artillery and heavy weapons to make a stand at Mullaittivu.
About 300,000 civilians are believed to live in the area.
Kilinochchi, the de facto capital of the Liberation Tigers of Tamil Eelam (LTTE), fell to government troops last week.
There has been no comment from the Tamil Tigers on the fall of the Jaffna peninsula, but communication with Tiger-held areas remains difficult and the government's claims are impossible to verify as journalists are barred from the region.
Tens of thousands of people have died since the LTTE began battling the government in 1983.
The LTTE says it is fighting for the rights of minority Tamils in the face of mistreatment by successive governments led by the Sinhalese majority since Sri Lanka won independence from Britain in 1948.