The bill had been stalled in the finely balanced parliament before Pauline Hanson, leader of the nationalist One Nation party, broke the impasse.
The latest Newspoll, published by The Australian newspaper on Monday, gave Turnbull a 40% approval rating.
Barnaby Joyce leads the rural-based National Party, the junior partner in the centre-right government led by Malcolm Turnbull's Liberal Party, a political alliance that has existed since 1923.
Malcolm Turnbull, whose coalition holds a razor-thin majority of just one seat, said on Thursday that Barnaby Joyce had shown a "shocking error of judgment" for conducting an affair.
Malcolm Turnbull, who lost his one-seat majority and was being called to step down before Christmas, said that he "runs the government" and that his party's policies would soon be converted into public support.
A citizenship crisis that robbed Australian Prime Minister Malcolm Turnbull of his one-seat majority will claim another politician, with the Senate president on Wednesday saying he will resign after confirming he was a dual national.
Australian Prime Minister Malcolm Turnbull on Saturday appointed his foreign minister as acting prime minister as he tried to stem the fallout of a citizenship crisis that has cost his government its parliamentary majority.
Prime Minister Malcolm Turnbull proposed the tougher rules in April, adding stringent English-language tests and a quadrupling of the residency requirement for citizenship applicants from one to four years.
Australia's widening citizenship crisis entangled a seventh politician on Friday, a key independent senator whose support is critical for Prime Minister Malcolm Turnbull to pass legislation through a hostile Senate.
US President Donald Trump was wasting his time deriding the media over their coverage of his administration, Australian Prime Minister Malcolm Turnbull said.
The Australian government approved a second international airport for Sydney on Monday, opening the way for construction to begin in Australia's largest city on a project mired in political squabbling for 70 years.