Prime Minister Abbott made the comment after touring the Pearce RAAF air base north of Perth, where planes from seven nations are being deployed to the Indian Ocean about 1850km west of the city.
"This is a major international incident and Australia has the lead responsibility, if you like, for operations inside our search and rescue zone," he told reporters on Monday.
Mr Abbott said each country involved in the search was currently bearing its own costs and Australia was paying for running the coordination centre, which will have about 20 staff and be led by retired air chief marshall Angus Houston from its Perth CBD headquarters.
"It's a cost that we think is only reasonable - as the country in whose search and rescue zone the aircraft has come down - it's only reasonable that we should bear this cost," he said.
"It's an act of international citizenship on Australia's part.
"At some point, there may need to be a reckoning, some kind of tallying, but nevertheless we are as happy to be as helpful as we can to all of the countries with a stake in this."
Mr Abbott said Australia was not putting a time limit on the search.
Deputy Prime Minister Warren Truss said there had been a surge in the momentum of the search, with more ships and aircraft being added to the international effort every day.
Mr Truss said the priority was to recover the black box as soon as possible.
The Australian navy vessel Ocean Shield, which is equipped with special equipment that can detect signals from the black box, is being sent to the search site.
It's a race against time, given the box's low frequency acoustic beacon has a battery life of roughly 30 days.
The plane was last heard from on March 8.
Mr Truss said the Australian Transport Safety Bureau had a special skill in being able to interpret data from the black box, which is an Australian invention.
Mr Abbott said it was heartening to see countries including China, Japan and South Korea working together to solve the mystery, while Malaysian aircraft had joined the southern Indian Ocean search in recent days after previously combing Asian waters.
"It has been tremendous to see the international co-operation here," he said.
"It demonstrates that in a humanitarian cause, the nations of this region can come together to work for the betterment of humanity, can work to try to resolve this extraordinary mystery, can work to try to bring peace and closure to the families of the 239 people on board that ill-fated aircraft."
Nonetheless, the operation remained a massive challenge, Mr Abbott said.
"This is an extraordinarily difficult exercise.
"We are searching a vast area of ocean and working with quite limited information."
While 550 personnel were operating from the Pearce air base and about 1000 sailors were looking for debris at sea, Defence Minister David Johnston indicated that the task was still onerous.
"We're doing our very best - bear with us," he said.
Mr Abbott also said his Malaysian counterpart was not too hasty in announcing last week - before any debris had been recovered or confirmed as being from MH370 - that the plane was lost in the southern Indian Ocean and all on board were assumed dead.
"The accumulation of evidence is that the aircraft has been lost and it has been lost somewhere in the south of the Indian Ocean - that's the absolute overwhelming weight of evidence and I think that Prime Minister Najib Razak was perfectly entitled to come to that conclusion.
"And I think once that conclusion was arrived at, it was his duty to make that conclusion public."
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