The Shiite militant group has sent thousands of fighters into neighbouring Syria to fight alongside the troops of President Bashar al-Assad.
Addressing supporters in Beirut's southern suburbs by a video link, Nasrallah said that Sunni radicals, known as takfiris, "have no future".
"These takfiris will be defeated in all areas and countries, and we will feel honoured that we played a role in their defeat," he said.
The Ashura commemorations, which mark the killing of Imam Hussein, took place amid tight security because of threats linked to Syria-related violence in Lebanon.
Hussein, the grandson of the Prophet Mohammed, was killed at the hands of soldiers of the caliph Yazid in 680 AD, an event that lies at the heart of Islam's sectarian divide into Shiite and Sunni sects.
Beirut's southern suburbs -- a Hezbollah stronghold -- have seen a string of deadly attacks, many of them claimed by jihadist groups, since the militant group started sending fighters to Syria three years ago.
"We are now in the fourth year (of the Syrian war), and the extremists have failed to take over Syria... it's a great victory," Nasrallah said.
"We want to win the final victory... so that the region does not fall into the hands of beheaders... and rapists," he said, referring to the Islamic State jihadist group, which has carried out widespread atrocities in Syria and Iraq.
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The Syrian conflict began as a peaceful movement demanding democratic change, but later morphed into a brutal civil war after Assad unleashed a massive crackdown on dissent.
Hezbollah says it is fighting to stop extremism from spreading into Lebanon.
Its involvement in the Syrian war has deepened rifts in Lebanon, where most Sunnis support the anti-Assad revolt, while most Shiites back the Assad regime.
Hezbollah and the Lebanese army locked down the southern Beirut suburbs because of fears of new violence during Ashura, allowing only pedestrians and buses to enter.
Most worshippers wore black clothes, while some wore green scarves.
Many thousands marched, while others sat in silence in the rain, weeping in memory of the slain imam.
"In the beginning, we used to say the United States and Israel (were the enemy), but now, there is a new enemy which is at least as dangerous as Israel," said 50-year-old Abdel Karim Mansour, referring to Sunni radicals who brand Shiites as heretics.
Fatima Raslan, a 59-year-old at the rally, said: "We will crush IS members under our feet. They won't be able to defeat us."
In the southern Lebanese city of Nabatiyeh, Hezbollah ally Amal also held a procession at which marchers cut themselves with knives in honour of Imam Hussein.