Tasmanian senator Jacqui Lambie has confirmed she will vote with the government to pass its $158bn tax cut package, handing the Coalition a significant victory in the first week of the new parliament.

Lambie’s vote gives the government the final of four crossbench Senate votes it needs to pass the 10-year plan without the support of Labor, which had been trying to win support to amend the package.

Speaking on Thursday morning ahead of the legislation being introduced to the Senate, Lambie said she had outlined her concerns about Tasmania’s public housing debt, and had been given an undertaking that the government would respond.

“Yes, I will be supporting the tax cuts,” she told ABC Radio National.

“Bottom line (is) now there’s deals on the table, where I can give businesses down there and give others a thousand dollars so I can stimulate that economy and so some people can pay their bills,” she said.

“I’m not keen, I’ve made that quite clear, on the stage three tax cut, (but) they are another five, six years away, and we have got an election in between.”

On Wednesday, Lambie had called on the Coalition to waive Tasmania’s social housing debt as a condition for her support of the $158bn tax plan, saying she could not support tax cuts when thousands of low-income earners in her home state were struggling with homelessness.

“There is no way in good conscience I can vote for substantial tax cuts without making sure that the people who so desperately need a roof over their heads aren’t left to go without,” Lambie said in a statement on Wednesday.

Lambie said the money she was asking for – about $15m a year – was “very little in the overall of things”, and she had been assured that the federal government would now act.

“The goodwill is there, they know we have got housing problems down there, we have got massive (numbers of) homeless people down there, and they want to do something as well,” she said.

“We are doing what we can as quickly as what we can.

“What I don’t want to be doing is rushing out, saying here’s the money and that’s it. We want to make sure that money is targeted.”

Lambie said that she trusted the government would waive the debt, which costs the state $15m in interest repayments a year, but more time was needed to “iron it out properly.”

With the support of Lambie, the two Centre Alliance senators and Cory Bernardi, the government has now secured the 39 votes it needs in the Senate, rendering the support of Labor unnecessary.

Labor MPs have been divided on whether to block the government’s tax plan, and the party has yet to declare how it will vote on the unamended legislation when it comes to the Senate on Thursday.

Finance minister Mathias Cormann welcomed Lambie’s support, and said the government had negotiated in good faith.

“As we have always said, we are always prepared to talk about policy issues of concern to our crossbench Senate colleagues, but of course all of these judgements have to be made on their own merit,” Cormann told Sky News.

“Of course we have agreed with those crossbench colleagues that we will continue to work with them through those issues.”

“Today is very good news for millions of working Australians who will get to keep more of their own money.”

Tasmanian senator Eric Abetz (second from left) says wiping Tasmania’s social housing debt ‘rewards bad behaviour’. Photograph: Sam Mooy/AAP

Both Greens and Labor had been lobbying the crossbench not to side with the government, with Labor hoping senators would support its proposed amendments to defer stage three of the tax package and bring forward tax cuts for people earning up to $120,000.

The Tasmanian Greens senator, Peter Whish-Wilson, said on Wednesday Lambie would be “selling out Tasmanians” if she sided with the Coalition.

The shadow education minister, Tanya Plibersek, said there was no reason to rush the third stage of the tax cuts, due to come into effect in 2024, because they were on the “never never”.

“We need the stimulus in the economy now, we support immediate tax cuts, we think immediate tax cuts should be bigger,” Plibersek told Sky News.

“What we don’t want to see is us signing a blank cheque now for revenue that we will have to find in years to come.”