Phil Glendenning, President of the Refugee Council of Australia said, "We thank and congratulate all the Members of Parliament and Senators who came together and voted with their conscience on this long overdue legislation".
Denying that he would try to overrule the vote by blocking it from getting royal assent, Morrison vowed to reverse the "foolish" law should the Coalition be reelected when the polls open in May.
"The people of Australia will remember this day and know that this is now on your head", the Prime Minister said to Mr Shorten in Parliament.
The move cut short a furious debate over whether the vote in the lower house could be seen as a vote on a money bill and was therefore a vote of no confidence in the government itself.
The bill, which must be approved by the Senate before it becomes law, would give a panel of two doctors the power to approve transfers from offshore detention.
It was the first time in nearly 80 years that a federal government had been rolled on significant legislation in the House of Representatives.
The Justice Party senator said it was the "toughest decision" of his time in parliament.
"Once a government loses control of government finances, it is obliged to resign or seek a dissolution", Professor Twomey told The Sydney Morning Herald and The Age.
For the bill to pass the Senate, where it is likely to return on Wednesday, 39 votes in favour will be needed.
Under the negotiated amendments, the home affairs minister will have 72 hours to make a decision on whether to agree to a medical transfer.
"With the last few children leaving Nauru and the provision of adequate medical care now in sight, we are hopefully witnessing an end to the race to the bottom on refugee policy", said Mr Glendenning.
The later version of the Labor proposal applies the same section of the Migration Act but adds the condition that the minister must "reasonably believe the person would expose the Australian community to a serious risk of criminal conduct".