The Palestinian presidency said Israel was "playing with fire". It comes at a time of particularly high tensions following months of deadly incidents.
The march celebrates Israel's capture of East Jerusalem in the 1967 war.
Israel regards the whole of Jerusalem as its capital, something rejected by most countries and the Palestinians.
The Palestinians claim Israeli-occupied East Jerusalem as the capital of a future state of their own - ruled out by Israel - and political displays by either side are viewed by the other as direct challenges.
The BBC's Yolande Knell said the scene was raucous by the Damascus Gate entrance of the Old City, as young Israelis chanted and danced, carrying flags and playing drums as they headed towards the Muslim Quarter.
They shouted out: "The people of Israel live!"
Some Jewish Israeli families were among the crowd, carrying pushchairs down the narrow steps. There was a heavy presence of Israeli police, some with dogs guarding over the proceedings, she said.
But in this space where there would be usually large numbers of Palestinians going about their business or sitting on the stairs at that time of day there were only a handful of Palestinian journalists, our correspondent added.
Earlier, there were violent confrontations between Palestinians and Israeli police at a flashpoint holy site in the Old City. Shortly after, hundreds of Jewish visitors, including a far-right MP, ascended the hilltop site, where some danced, waved Israeli flags and bowed down to pray, before being stopped by police. Palestinians view such actions as incendiary and militant groups had warned they would not tolerate it.
"Israel is irresponsibly and recklessly playing with fire by allowing settlers to desecrate the holy sites" in East Jerusalem, the president's spokesman Nabil Abu Rudeineh said, according to the official Palestinian news agency Wafa. Palestinian officials often describe Israelis who visit the holy site revered by Muslims and Jews as settlers and their presence there as a desecration.
Jews and other non-Muslims are allowed to visit the site at certain times but are forbidden to pray or display any religious or national symbols under a long-standing agreement with the Muslim authority which administers the compound.
