"I am confident that the enemy will run away in degradation and embarrassment," Omar said.
"Afghans have always expelled their enemies by force and no enemy or aggressive force has left Afghanistan at its own will... They have committed aggression and we will drive them out."
Omar also said the Eid festival, which began on Saturday, offered an example of sacrifice which could inspire his fighters.
An ideology based on sacrifice, such as the Taliban's, "never submits and accepts defeat", he said.
"You can see that present resistance and struggle in Afghanistan has amply proved the point."
Omar's Taliban movement governed large parts of Afghanistan before being defeated by the Northern Alliance and a coalition of international troops in 2001.
Omar denounces government plans
In the tape, Omar also criticised a new plan by the Afghan government to hold tribal councils (jirgas) on both sides of the Afghan-Pakistani border in an effort to find ways to end recent violence.
|The Taliban leader has told his followers to avoid killing Afghan civilians [AP]|
"Now the aggressor forces in our country want to entangle our valiant nation and tribes in their devilish trap by way of jirgas," he said.
"But I am sure that no Muslim will participate in something that is created by the aggressors and puppets. Those who attend will only be people who have sold out."
"Our aggressor enemy has been defeated and now they are hatching new conspiracies for their survival," he said.
Afghanistan and Pakistan are still considering how the jirgas can be organised and who will take part.
Taliban urged to focus attacks
Omar also instructed his fighters to be "mindful of not resorting to actions that may result in casualties of innocent ordinary people".
"We must be more cautious and careful in focusing the target. We should have friendly and sincere relations with our own Muslim people."
His advice echoes similar statements by al-Qaeda leaders who are concerned that indiscriminate attacks by Islamist groups have undermined Muslim support for their causes.
More than 4,000 people have been killed in Afghanistan this year, the bloodiest since the Northern Alliance ousted the Taliban government in 2001.