BAGHDAD—Iraq’s new prime minister was sworn in late on Wednesday after lawmakers approved a majority of his cabinet, overcoming the first test of his leadership and giving the country a government five months after elections.

The parliament approved 14 of the 22 ministers nominated by Prime Minister Adel Abdul-Mahdi following a tempestuous, four-hour session that some lawmakers threatened to disrupt saying they needed longer to vet the nominees because some were suspected of corruption or may have fake degrees. Several key...