The amendment was introduced by the senate on Friday and then passed by the lower house on Sunday.
The original version of the bill said a census was needed before elections, effectively delaying next year's polls.
The opposition said this was an attempt to extend President Joseph Kabila's time in office.
Under the new bill, the election can proceed as planned, without a census, which would take at least a year in this vast country.
Human rights groups say that dozens of people were killed during the protests. The government puts the figure at 12 - a policeman shot by a sniper and 11 looters killed by security guards.
"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 on Friday.
The lower house, the House of Representatives, had approved the original version of the bill, in a vote boycotted by opposition MPs.
The government has argued that the census is vital to ensure polls are free and fair - the country has not had a reliable census since independence from Belgium in 1960.
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 Democratic Republic of Congo parliament has passed an amended census bill following four days of violent nationwide protests.