The government will no longer need legislation to make changes to welfare settings after passing an amendment to the package, giving the social services minister unprecedented powers.
Greens Senator Mehreen Faruqi, who had pressed the government to extend the payment to students, said it was important to fight to protect them throughout this crisis.
"At a time when students are losing work and their studies are up in the air, they are absolutely deserved of the same support as anyone else," she said.
But the payment is not available to international students.
In a statement, the Innovative Research Universities (IRU) network voiced its concern for this group.
"The IRU remains concerned that many international students, who are not eligible for the coronavirus supplement, may struggle to make ends meet during the current crisis, particularly as international students have few support options if they lose their part-time jobs," it said.
Parliament has passed two packages worth $17.6 billion and $66 billion in response to the outbreak of the virus.
In separate legislation, the government set aside a further $40 billion for urgent and unforeseen spending associated with the pandemic, which is likely to cause a recession.
Jobseeker, youth allowance, parenting and special benefit payments will be boosted by $550 a fortnight.
There'll also be two $750 payments for welfare recipients.
Prime Minister Scott Morrison addressed a scaled-down House of Representatives on Monday, invoking the spirit of the Diggers.
"So we summon the spirit of the Anzacs, of our Great Depression generation, of those who built the Snowy, of those who won the great peace of World War II and defended Australia," Mr Morrison said.
"For many, young and old, 2020 will be the toughest year of our lives."
Labor leader Anthony Albanese said most Australians only knew stories of war, hunger and financial strife from their grandparents.
"This is a time for national leadership, consistent messages, clear directions," he said.
Treasurer Josh Frydenberg has hinted at the potential need for a third stimulus package down the track, as Australia faced its biggest challenge outside of war time.
"We confront an enemy without a flag or a face and we are deploying every weapon in our arsenal to defeat it," he told parliament.
"We will bounce back as a nation stronger than ever."