The incident is prompting the 39-year-old Labour MP to reconsider her political career, coming after a spell of publicly-documented poor mental health.
On Sunday night, Ms Allan was held into police custody for around two hours after a car crash in the Wellington suburb of Roseneath.
Radio NZ reports she crashed into a parked car.
A sombre Prime Minister Chris Hipkins fronted media on Monday morning, saying Ms Allan is "clearly not in a good space" after accepting her resignation.
"Everyone's still processing this, including me," he said.
"I'm sad for Kiri who is an incredibly talented person, who clearly has been battling some demons."
In a statement, Ms Allan said she would immediately resign.
"I'm heading home and will be taking time to consider my future in politics," she said.
"I accept that my position as a minister is untenable.
"I'm very sorry for my actions, the harm they could have caused and the embarrassment it has placed on the government and my colleagues."
The scandal is the latest in a long line of problems for Mr Hipkins, and in all likelihood ends the political career of Ms Allan, once a Labour rising star.
Promoted to cabinet after the party's 2020 landslide win, Ms Allan took leave in 2021 after being diagnosed with stage three cervical cancer.
She declared herself cancer-free and returned to the ministry later that year.
In recent months, she was reprimanded for criticising Radio NZ chiefs at an informal function, and accused of bullying public servants in her office.
Ms Allan apologised and spoke of her difficulties with mental health during a personal relationship breakdown, with Mr Hipkins giving her time off and requiring her to undertake counselling.
The car crash comes a fortnight after that apology.
"Over recent weeks I've faced a number of personal difficulties. I took time off to address those, and believed I was okay to juggle those challenges with the pressure of being a minister," she said.
"My actions yesterday show I wasn't okay, and I've let myself and my colleagues down."
Mr Hipkins said Ms Allan's actions were "inexcusable" but occurred as "she was experiencing extreme emotional distress".
"Last night's incident is evidence there are not always quick fixes when it comes to mental health," he said.
Ms Allan's resignation means Labour's previously 20-strong cabinet is down to 18 members.
Transport Minister Michael Wood resigned last month after failing to declare shareholdings relating to his ministries, despite a dozen requests from the cabinet office to do so.
In a scandal-plagued year, Labour has also lost Customs Minister Meka Whaitiri after she defected to the Maori Party, and Police Minister Stuart Nash, for revealing cabinet discussions to donors.
Labour has now lost nine ministers from the team that started the year, including prime minister Dame Jacinda Ardern.
Mr Hipkins, who replaced Dame Jacinda as prime minister in January, faces an uphill battle to convince voters Labour deserves a third term at the October 14 election.
The prime minister will announce a fresh ministerial reshuffle later on Monday, but confirmed no new additions to cabinet.
"We've got five sitting weeks to go until the election ... I don't think now is the right time to be bringing new people in," he said.
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