Mexico’s Senate on Wednesday narrowly passed a sweeping proposal to revamp the judiciary system, effectively clearing the last major obstacle to a measure that the country’s president had vowed to push through before stepping down at the end of this month.

The result reflects the exceptional sway of President Andrés Manuel López Obrador of Mexico and his party after his allies won large legislative majorities in June, enabling them to pass some of the Mexican leader’s most contentious and far-reaching proposals in his final weeks in office.

The measure would shift the judiciary from an appointment-based system largely grounded in training and qualifications to one where voters elect judges and there are few requirements to run — and it would remove 7,000 judges from their jobs, from the chief justice of the Supreme Court down to those at local courts.

The bill already passed in the lower house of Congress last week, during a marathon session. It will now go to the state legislatures, where it will need a majority to be enacted into law. Mr. López Obrador’s governing Morena party and its allies control 25 of 32 state legislatures, so it is expected to be approved with ease.

When that happens, voters could start electing thousands of federal, state and local judges as soon as next year.

The debate, which started on Tuesday, was temporarily suspended after a group of protesters, megaphones and Mexican flags in hand, barged into the Senate building calling on senators to block the overhaul. Protesters followed the lawmakers to another venue, where an opposition senator was assaulted when someone threw gasoline on his face. Police officers later dispersed demonstrations using fire extinguishers.