The International Monetary Fund (IMF) has confirmed that Greece has cleared overdue debt repayments of €2.05bn (£1.4bn) and is no longer in arrears.
The repayments, and another for €4.2bn to the European Central Bank (ECB) due on Monday, came after the EU made Greece a short-term loan of €7bn.
Cash-strapped Greece missed one repayment to the IMF in June and another earlier this month.
Earlier on Monday, Greek banks reopened after being closed for three weeks.
However, many restrictions remain and Greeks are facing price rises with an increase in Value Added Tax (VAT).
Analysis by Robert Peston, BBC economics editor
Just because the doors of Greek banks are open today, don’t be fooled into thinking they and the Greek economy are anywhere near back to recovery.
There are still major restrictions on the ability of their customers to obtain their cash or move it around.
The symbolic importance of the European Central Bank turning on the emergency lending tap again was important, but it has only been turned on a fraction.
It has given enough additional Emergency Liquidity Assistance – €900m – to keep the banks alive in a technical sense.
There is no possibility of them thriving for months and even possibly years.
IMF spokesman Gerry Rice confirmed in a statement that Greece had repaid the totality of its arrears.
“As we have said, the fund stands ready to continue assisting Greece in its efforts to return to financial stability and growth,” he said.
Greece missed its first repayment to the IMF on 30 June and another on 13 July during deadlock over negotiations for a third bailout.
The crisis brought Greece to the brink of economic collapse and an exit from the euro.
The government has since reached a cash-for-reforms deal with its creditors and negotiations are due to begin on the proposed €86bn rescue package.
For the past three weeks, Greeks have been waiting in line at cash machines to withdraw a maximum of €60 (£41) a day, a restriction imposed amid fear of a run on the banks.
From Monday, the daily limit becomes a weekly one capped at €420 (£291), meaning Greeks will not have to queue every day.
However, a block on transfers to foreign banks and a ban on cashing cheques remain in place.
VAT is rising from 13% to 23% meaning Greeks will pay more on a range of goods and services, including taxis and restaurants.
The rise was among a package of reforms demanded by Greece’s creditors.
Dimitris Chronis, an Athens kebab shop owner, said the new taxes were bad news for his business.
“I can’t put up my prices because I’ll have no customers at all,” he said.
“We used to deliver to offices nearby but most of them have closed. People would order a lot and buy food for their colleagues on special occasions. That era is over.”
Prime Minister Alexis Tsipras faced a rebellion from within his left-wing Syriza party over the tough austerity measures being demanded by other eurozone leaders, who are among Greece’s creditors.
But last week’s vote in the Greek parliament paved the way for Greece to receive the €7bn bridging loan that enabled the reopening of the banks.
Mr Tsipras has since replaced his rebel ministers but analysts say his government has been weakened and fresh elections may be held in September or October.
The Greek parliament is due to hold a second vote on Wednesday on measures including justice and banking reforms. The government is again likely to scrape through, supported by opposition parties.
Representatives from Greece’s creditors – known as the Troika – are due to arrive in the country soon and talks on the new bailout are expected to last about a month.
The tough negotiations over Greece have also revealed divisions within the eurozone about the future of the bloc.
Germany, which is the largest contributor to Greek rescue funds, has taken a tough line on Greece, while other states, such as France, have appeared more conciliatory.
On Monday, French President Francois Hollande put forward his ideas for a new parliament for the eurozone countries and a shared budget.
The eurozone is currently managed by the Eurogroup, made up of the finance ministers of each nation.