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".
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The prime minister had judged this to be "potentially helpful to him in reaching an agreement", he noted, adding: "I shall wear the creditors' loathing with pride."
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.
The damoclean sword hanging over Greece is that the banks will run out of cash in just a few days.
Which means that a decision on what kind of limited service the banks can provide, to be made by the Greek government, will hinge entirely on whether the European Central Bank's (ECB) governing council decides to freeze, cut or increase the provision of Emergency Liquidity Assistance, or emergency loans to Greece.
As for the ECB, it does not wish to be seen as Greece's Judge Dredd. It will take its lead from any statement by eurozone government heads on whether there is a realistic chance of a new deal to rescue the finances of the ailing Greek state.
So the fate of Greece lies firmly where it always did, in the lap of the German Chancellor, Angela Merkel and her fellow heads of government.
And here is the horrendous analysis of costs that they need to make.
Robert Peston: Huge costs of Greece staying in or quitting euro
Will 'No' vote hasten Greece's exit from the euro?
Gavin Hewitt: Greece votes 'no' and gambles future
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.
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.
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.