Government forces have advanced towards the final remaining strongholds of the Liberation Tigers of Tamil Eelam (LTTE) since the Tigers' headquarters at Kilinochchi fell on Friday.
Troops were also close to taking a key land route to the northern peninsula of Jaffna, and have made progress in securing the strategic Elephant Pass, the ministry said.
The Tamil Tigers have vowed to hit back after suffering a series of major defeats in their vicious 37-year battle for a separate homeland.
The ministry said troops recovered three rebel bodies while "scores" were killed during heavy fighting around the Muhamalai defence line in the Jaffna peninsula on Tuesday.
"Security forces... are now gaining rapid tactical advantage leaving LTTE near to extinction," the ministry said, adding that intense ground battles had left "well-fortified LTTE defences crumbling and