The final result in the referendum was 61.3% “No”, against 38.7% who voted “Yes”. Turnout was 62.5%.
Thousands of Greeks celebrated in the streets after the vote, waving flags, chanting and setting off fireworks.
But European officials and “Yes” supporters have warned the result could see Greece ejected from the eurozone.
The euro fell across the board in Asian markets after the referendum.
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”.