Prime Minister Alexis Tsipras has said Greeks made a “brave choice” in voting to reject the terms of an international bailout in Sunday’s referendum.
Thousands celebrated in the streets after hearing the final result was 61.3% “No”, against 38.7% “Yes”.
But European officials warned that it could see the country ejected from the eurozone and the euro fell across the board in Asian markets on Monday.
Greece’s finance minister, who often clashed with creditors, has resigned.
Yanis Varoufakis wrote on his blog that he had been “made aware of a certain preference by some Eurogroup participants, and assorted ‘partners’, for my… ‘absence’ from its meetings”.
The prime minister had judged this to be “potentially helpful to him in reaching an agreement”, he added.
Greece’s governing Syriza party had campaigned for a “No”, saying that the bailout terms were humiliating.
Mr Tsipras said late on Sunday that the Greeks had proved that “democracy won’t be blackmailed.”
Speaking in a televised address, he said: “Given the unfavourable conditions last week, you have made a very brave choice.”
“But I am aware that the mandate you gave me is not a mandate for rupture.”
He said that Greece would go back to the negotiating table on Monday, adding that an International Monetary Fund (IMF) assessment published this week confirmed that restructuring Greek debt was necessary.
But some European officials had already warned that creditors could take a “No” vote to mean that Greeks had rejected further talks.
European leaders were varied in their openness to more talks with Greece after the vote.
Germany’s Deputy Chancellor, Sigmar Gabriel, told local media that renewed negotiations with Greece were “difficult to imagine” and that Mr Tsipras had “torn down the bridges” between Greece and Europe.
However, Italian and Belgian ministers were more conciliatory, with Belgium’s finance minister saying that the door remained open to restart talks “literally, within hours”.
Jeroen Dijsselbloem, who heads the eurozone’s group of finance ministers, said the referendum result was “very regrettable” for the future of Greece.
Meanwhile, Martin Schulz, the president of the European Parliament, warned that Europe was entering “a very difficult and even dramatic time” unless the Greek government made “meaningful” proposals in the coming hours.
Greece had been locked in negotiations with its creditors for months when the Greek government unexpectedly called a referendum on the terms it was being offered.
Robert Peston, BBC economics editor, Athens
Greek banks are desperately in need of a lender of last resort to save them, and the Greek economy.
And sad to say no banker or central banker to whom I have spoken believes the European Central Bank (ECB) can fulfil that function – because it is struggling to prove to itself that Greek banks have adequate assets to pledge to it as security for new loans.
There are only two options. The Bank of Greece could make unsecured loans to Greek banks without the ECB’s permission – which would provoke a furious reaction from Eurozone leaders and would be seen by most of them as tantamount to leaving the euro.
Or it can explicitly create a new currency, a new drachma, which it could then use to provide vital finance to Greek banks and the Greek economy.
Banks have been shut since last Monday after the European Central Bank declined to give Greece more emergency funding.
Withdrawals at cash machines have been limited to €60 per day. Greece’s last bailout expired on Tuesday and Greece missed a €1.6bn (£1.1bn) payment to the IMF.
More European reaction:
• Italian Foreign Minister Paolo Gentiloni said: “Now it is right to start trying for an agreement again. But there is no escape from the Greek labyrinth with a Europe that’s weak and isn’t growing.”
• UK eurosceptic party leader Nigel Farage welcomed the result. He tweeted: “EU project is now dying. It’s fantastic to see the courage of the Greek people in the face of political and economic bullying from Brussels.”
• EU foreign policy chief Federica Mogherini tweeted that the vote meant “Painful days for all those that believe in a united Europe”.
• The leader of Spain’s leftist anti-corruption party Podemos, Pablo Iglesias, was more positive. “Today in Greece democracy won,” he said.
Greek government officials have insisted that rejecting the bailout terms would help them negotiate a better deal with international creditors.
Mr Varoufakis wrote that the “No” referendum vote was a “majestic, big yes to a democratic Europe”. Earlier, he said that banks would reopen on Tuesday.
The coming days will see a flurry of emergency meetings across Europe.
European Commission President Jean-Claude Juncker said he would have a conference call with key EU officials and the ECB on Monday morning.
French President Francois Hollande and German Chancellor Angela Merkel are scheduled to meet in Paris on Monday. A summit of eurozone heads of state has been arranged for Tuesday.
The European Commission – one of the “troika” of creditors along with the IMF and the ECB – wanted Athens to raise taxes and slash welfare spending to meet its debt obligations.
Greece’s Syriza-led government, which was elected in January on an anti-austerity platform, said creditors had tried to use fear to put pressure on Greeks.