The vote in favour of the Bill came amid growing concern about the well-being of asylum seekers sent to Nauru and Papua New Guinea, with reports of abuse, suicide and lengthy detention periods.
The Labor caucus agreed on Monday night to authorise negotiations on the crossbench bill to widen the definition of security in the bill to allow refugees to be rejected if they have a criminal history, to give the Immigration Minister more than 24 hours to decide transfers and to limit the rules to about 1,000 individuals now on Nauru and Manus Island.
"That would be deeply irresponsible and I would hope they don't do that", he told reporters.
Mr Morrison ruled out calling a snap election, saying the vote was not a no-confidence motion in his government and he was still planning for a national poll in May.
The last time a government lost a vote on legislation on the floor of the house was in 1929, while in 1941 the government lost a vote on a budget bill.
During debate on Labor's amendments to the bill, Finance Minister Mathias Cormann told the Senate that "rapists and paedophiles will still get a free pass into this country".
Senator Hinch backed the bill in the Senate late a year ago.
Australian security agencies warned in December that if those amendments became law, asylum seekers would likely head to Australia again in significant numbers.
The bill must be approved by the upper house to become law, but is likely to pass, as the Senate supported an earlier version previous year.
Morrison said he would reopen the detention centre, which was closed a year ago, to cope with what he predicted would be an influx of transfers from Manus Island and Nauru.
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Labor is accusing the government of a hysterical response.
"I believe we can keep our borders secure, we can uphold national security, but still treat people humanely", Mr Shorten told parliament.
The Greens are concerned about Senate delays and about the way the amendments give the Immigration Minister more discretion, which could force applications for medical transfers to go through the courts as they are today. Rape survivors have had to have traumatic late term abortions due to government blocks.
Labor representatives said it was about "doing the right thing", and the vote provoked cheers from asylum-seeker activists in parliament's public gallery.
"We are doing whatever it takes to prevent this bill from becoming law and you will of course see how these processes are going to evolve over the next 24 to 48 hours", he said.
Also in the Swiss city on Wednesday night, Abdul Aziz Muhamat, a Sudanese refugee activist who has spent five years on Manus Island, won the Martin Ennals Award 2019, one of the most prestigious global human rights prizes, organizers said.
The centre on Australian territory has been the scene of violent riots, hunger strikes and numerous incidents of self-harm.
In 2010, a boat of 90 asylum seekers - mostly from Iraq and Iran - sank off the island, killing 50 people on board.
Because it was the most prison-like of Australia's immigration centres, Christmas Island had been repurposed in recently years to house foreign criminals who had completed sentences in Australian jails and were fighting deportation.
Home Affairs Minister Peter Dutton said part of the Christmas Island facilities "can be opened very quickly". "This puts Australia back on the map for people smugglers and Bill Shorten has that on his shoulders".
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