Khamenei chanted "Death to Israel" and "Death to America".
"In the coming days the world will witness Iran's announcement of its very important and very major nuclear achievements," Ahmadinejad told a crowd at Tehran's Azadi (Freedom) Square in a speech relayed live on state television.
He gave no details.
"The world should know that despite all pressures, Iran will not withdraw one inch from its principles and (nuclear) rights," he said.
"All countries have put pressure on us for not obtaining nuclear knowhow, but all these pressures were futile. We not only now have the nuclear knowhow but also are capable of providing for our (nuclear) needs by our own local experts," Ahmadinejad said.
The ISNA news agency quoted Ahmadinejad as telling reporters at the anniversary rally that "Iran would definitely make history and change the world."
Later in his speech, Ahmadinejad called on world powers to stop using threats and instead return to negotiations based on mutual respect and justice.
"We are ready for resuming the nuclear talks but you (world powers) come up every day with new excuses not to the resume the talks," he said.
"I tell you on behalf of the Iranian nation that any other option than returning to the negotation table would be doomed to fail," Ahmadinejad added.
Tension with the West over Iran's disputed nuclear work has risen in recent weeks. The United States and its European allies have imposed new sanctions to try to force Iran back to talks before it produces enough nuclear material for an atomic bomb.
Iran says its nuclear program has only peaceful purposes.
The U.S. and Israel have not ruled out military action against Iran to prevent it from obtaining nuclear weapons.
Iran has warned of a "painful" answer, saying it would hit Israel and U.S. bases in the Gulf as well as block the vital Gulf oil shipping route through the Strait of Hormuz.
"If attacked by the Zionist regime (Israel), we will turn it to dust," said a Revolutionary Guards commander, Mohammad Shirdel, semi-official Fars news agency reported on Saturday.
"Thousands of our missiles will target Israel and the 40 bases of America in the region," he added.
