The warning from Mullah Mohammad Omar came through an online Ramadan message, posted on Sunday to mark Eid el-Fitr, the end of the Islamic fasting month of Ramadan.
"It is the fifth consecutive year that Afghanistan, our dear country, is under Crusader colonisation," Mullah Omar said.
"But this time, we also congratulate you on the defeat and the flight of the Crusader.
"I also announce to you that [Karzai], the Crusaders' valet, and his colleagues are in the process of looking for a way to escape ... We are going to bring them before an Islamic tribunal," the Taliban leader wrote in Arabic.
"The Crusaders are leading people into error ... They are using all media means to promote their lies, spending $100 million.
"It is what the Russians did ... but I am assured that they will be defeated, just as the Russians and their allies were," the message said.
The Soviet Union occupied Afghanistan from 1979 to 1989.
But its troops were ultimately forced to withdraw after a struggle with Islamic forces that took the lives of least 10,000 Soviet soldiers according to official figures.
On October 7, a man identifying himself as Abdul-Hai Mutmaen and speaking in the name of Mullah Omar said that the Taliban spiritual head was alive and in Afghanistan, where he is directing an insurrection against the government and western troops.