"Greece leaves behind catastrophic austerity, it leaves behind fear and authoritarianism, it leaves behind five years of humiliation and suffering," Tsipras told thousands of cheering supporters gathered in Athens.
Syriza's victory marks a flat rejection of the model for troubled euro zone economies championed by German Chancellor Angela Merkel. It is also likely to strengthen calls for the euro zone to move towards policies which promote economic growth, rather that tackling budget deficits.
Syriza's campaign slogan "Hope is coming!" resonated with voters worn down by huge budget cuts and heavy tax rises during the years of crisis that have sent unemployment over 25 percent and pushed millions into poverty.
"We hope our expectations will be fulfilled," said 47-year-old teacher Efi Avgoustakou. "On Monday in class, we're not allowed to comment and take sides but we will be smiling."
Financial markets reacted nervously to the victory of Tspiras, who has promised to renegotiate Greece's debt agreements, fearing potential conflict with other euro zone governments that could put more strain on the currency bloc.
The euro slid to near an 11-year low and U.S. stock futures fell as Asian markets opened on Monday.
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Tsipras said he would cooperate with fellow euro zone leaders for "a fair and mutually beneficial solution" but said the Greek people came first. "Our foremost priority is that our country and our people regain their lost dignity," he said.
He has promised to keep Greece in the euro and toned down some of his rhetoric, but his arrival in power is likely to encourage other anti-austerity parties which are winning support across Europe, such as the Podemos movement in Spain.
It might also strengthen the hand of mainstream leaders including French President Francois Hollande and Italian Prime Minister Matteo Renzi who argue that orthodox austerity policies have failed to produce the economic growth which Europe needs to recover fully from the global financial crisis.
Hollande expressed in a statement his "desire to pursue the close cooperation between our two countries in service of growth and the stability of the euro zone".
Finnish Foreign Minister Erkki Tuomioja was more forthright, saying he believed the result would change the debate in Europe and put more emphasis on growth and employment. "This is a slap at what I see as a very right-wing economic policy in Europe," Tuomioja, a Social Democrat, told the website of the Helsingin Sanomat newspaper.
However, with Greece's economy unlikely to see any quick recovery from its crisis, Tsipras faces enormous problems and his victory raises the prospect of tough negotiations with European partners including Merkel.
Greece's bailout deal with the euro zone is due to end on Feb. 28 and Tsipras' immediate challenge will be to settle doubts over the next instalment of more than 7 billion euros in international aid. EU finance ministers are due to discuss the issue in Brussels on Monday.
One Syriza official expressed confidence that Tsipras could form a government by Wednesday. .