The protests were biggest in Kinshasa and the eastern city of Goma
The Democratic Republic of Congo senate has amended a controversial census bill following four days of violent nationwide protests.
The new version, to be approved by the lower house, removes the requirement to hold a census before the next election.
The opposition had said this was a way for President Joseph Kabila to extend his time in office rather than stand down next year as planned.
Hundreds of students took to the streets to celebrate the news.
Human rights groups say that dozens of people have been killed during the protests. The government puts the figure at 12 - a policeman shot by a sniper and 11 looters killed by security guards.
There were violent protests in Kinshasa and the eastern city of Goma
"We have listened to the street. That is why the vote today is a historic vote," Senate President Leon Kenga Wa Dondo said after the amendment was passed.
The BBC's Maud Jullien in the capital, Kinshasa, says it is uncommon for the senate to vote against the government in this way - the last time was 2010.
At the scene: Maud Jullien, BBC Africa, Kinshasa
As soon as the senate pronounced the vote I got a phone call from a student at the University of Kinshasa who had spent four days protesting. I could barely hear him - he was shouting over the phone and told me there were hundreds of students celebrating with him on the street. "We won," he told me.
The senate's vote reflected the riots that rocked many of the country's cities, especially the capital, where dozens are thought to have been killed when the police fired live bullets at protesters.
During the debate, senators said the bill, as it had been passed by the lower house of parliament, risked destabilising the country.
The senators' vote for an amendment of the controversial article is a victory for many of the protesters but it is not over yet - the bill still has to return to the House of Representatives before it can become a law.
The opposition says it would take about three years to hold a census in DR Congo, a country two-thirds of the size of western Europe, which has very little infrastructure and is home to numerous armed groups in the east.
The government has argued that the census is vital to ensure polls are free and fair - the country has never had a reliable census since independence from Belgium in 1960.
The university in Kinshasa saw some of the most serious protests
Under the senate's amendment, the elections could be held in 2016 as planned, before a census is conducted.
Mr Kabila first took power in 2001 following the assassination of his father Laurent, who was president at the time, and is constitutionally barred from running for another term.
The House of Representatives approved the plan for a census on Saturday, in a vote boycotted by opposition MPs.
The opposition says this amounts to a "constitutional coup" by Mr Kabila.