Al-Aqsa, Islam's third holiest site, and the Hurva are about just 700 metres apart.
The ceremony was attended by Reuven Rivlin, the Israeli parliament speaker, ministers and the country's chief rabbis.
In a video message to the ceremony, where Israeli politicians and chief rabbis were in attendance, Binyamin Netanyahu, the prime minister, said the Hurva brought a message of coexistence.
"We permit believers of other faiths to conserve their places of worship. We proudly protect our heritage, while at the same time allowing others freedom of religion," he said.
Khaled Meshaal, the exiled leader of Hamas which rules the Gaza Strip, denounced the synagogue's opening.
"We warn against this action by the Zionist enemy to rebuild and dedicate the Hurva synagogue. It signifies the destruction of al-Aqsa mosque and the building of the temple," he said from the Syrian capital, Damascus.
The al-Aqsa site is revered by Muslims as al-Haram al-Sharif (the Noble Sanctuary), comprising al-Aqsa mosque and the Dome of the Rock. It is known to Jews as the Temple Mount.
In a statement, issued at a meeting of leaders of Palestinian groups, Meshaal called the ceremony "a falsification of history and Jerusalem's religious and historic monuments".
"Israel is playing with fire and touching off the first spark to make the region explode," he said.
Earlier, Hatem Abdel Qader, the official in charge of Jerusalem affairs for Fatah, the party led by Mahmoud Abbas, the Palestinian president, said: "This is no mere synagogue.
"This synagogue will be a prelude to violence and religious fanaticism and extremism, and this is not limited to Jewish extremists but includes members of the Israeli government."
Al Jazeera's Sherine Tadros, reporting from the Hurva inauguration ceremony, said thousands of police and border guards were deployed across the Old City.
"Certainly the situation is tense here," she said.
"We are about 350 metres from al-Haram al-Sharif, one of Islam's holiest sites, and the towering presence of the Hurva synagogue has been called a provocation by Palestinian leaders and religious figures."
An Israeli government decision to include two West Bank religious sites in a Jewish national heritage plan has already angered Palestinians and raised tensions in recent weeks.
The announcement last week of Israeli plans for new settler homes near East Jerusalem has also contributed to the unrest.
Against this backdrop of tensions, Israeli soldiers injured 10 Palestinians on Monday in clashes with dozens of students hurling stones at a West Bank checkpoint, Palestinian medics and witnesses said.
"We have received six people, two of them wounded by live bullets, one in the stomach and the other in the neck," Mohammed Eida, the director of Ramallah area hospitals, told the AFP news agency.
The Palestinians had marched to the checkpoint from the nearby Birzeit University. Several of them threw rocks at the soldiers.
An Israeli military spokesperson said security forces responded by using riot-dispersal means and that there was no live fire.
Citing security concerns, Israel imposed a temporary closure of the West Bank on Friday.
Palestinians who do not carry Jerusalem residency have been banned from crossing into the city from the West Bank until Tuesday.
Men under the age of 50 have also been denied entry to al-Aqsa in the Old City since Friday.