Army spokesman, Major Shaban Bantariza, said that the operation launched earlier this year had been successful and that more than half of the 3,000 rebel troops had been killed.
Uganda's 'victory'
He dismissed suggestions that the timing of the announcement had anything to do with the fact that the commander of the army, Major-General James Kazini, said he would resign if the LRA was not wiped out by the end of the year.
The LRA has been conducting a campaign of violence in northern Uganda since 1987 to overthrow the government of President Yoweri Museveni.
'Skirmishes'
Major Bantariza said that Operation Iron Fist, design to flush out the rebels from their bases in southern Sudan, was now in its third phase and that it would soon be over.
"The war has been won," he told the BBC's Focus on Africa programme.
"It's not a general insecurity situation," he said.
He said that a majority of areas in the north had been made safe, and that there were "only a few skirmishes" which he described as "mop-up operations".
After the Ugandan army first entered southern Sudan in March, the LRA moved back into northern Uganda and intensified its attacks.
Last month, the Ugandan army offered a reward for information leading to the capture or killing of the chief of the LRA, Joseph Kony.
The army said it would pay $11,000 for information on the whereabouts of the leader of the rebellion.
It also put a price on the heads of the top commanders of the rebel group, an estimated 38 of them, which is feared for its brutality and cruelty.
Joseph Kony and his commanders are believed to be hiding in southern Sudan and northern Uganda.
But the army statement says a three-year-old amnesty offer is still open to the rebel leaders.
Biblical rule
Major Bantariza said that there was no reason why Major-General Kazini should resign.
"More than 90% of his mission has been achieved... Is 90% a failure, or not?" he asked.
"I'm sure the commander-in-chief [President Yoweri Museveni] is as satisfied with James Kazini's performance as the rest of us are," he said.
In November, Sudan agreed to let Ugandan soldiers remain on its territory, but the Ugandan army has been unable to stop LRA attacks.
The LRA has been demanding that Uganda be ruled according to the Biblical Ten Commandments.
But it has kidnapped thousands of children for use as fighters or sex slaves.