ROME—Parliament approved plans to cut the number of lawmakers by more than a third, a victory for Italy’s antiestablishment 5 Star Movement and a move that eliminates a potential source of friction in Prime Minister Giuseppe Conte’s new government.

The 5 Star, which has long railed at what it says is a bloated legislature, had made the move a central condition for forming a new government with the center-left Democratic Party in September after the leader of the far-right League, Matteo Salvini, pulled out of the previous...