With about 60 percent of votes counted, Syriza was set to win 149 seats in the 300 seat parliament, with 36.1 percent of the vote, around eight points ahead of the conservative New Democracy party of Prime Minister Antonis Samaras.
While a final result may not come for hours, the 40-year-old Tsipras is on course to become prime minister of the first euro zone government openly opposed to the kind of crippling austerity policies which the European Union and International Monetary Fund imposed on Greece as a condition of its bailout.
"Greece leaves behinds catastrophic austerity, it leaves behind fear and authoritarianism, it leaves behind five years of humiliation and anguish," Tsipras told thousands of cheering supporters gathered in Athens.
European leaders have said Greece must respect the terms of its 240 billion euro bailout deal, but Tsipras campaigned on a promise to renegotiate the country's huge debt, raising the possibility of a major conflict with euro zone partners.
Tsipras said on Sunday he would cooperate with fellow euro zone leaders for "a fair and mutually beneficial solution" but said the Greek people came first. "Our priority from the very first day will be to deal with the big wounds left by the crisis," he said. "Our foremost priority is that our country and our people regain their lost dignity."
Tsipras's campaign slogan "Hope is coming!" resonated with voters worn down by huge budget cuts and heavy tax rises during six years of crisis that has sent unemployment over 25 percent and pushed millions into poverty.
With Greece's economy unlikely to recover for years, he faces enormous problems and his victory raises the prospect of tough negotiations with European partners including German Chancellor Angela Merkel.
As thousands of flag-waving supporters hit the streets of Athens, some shedding tears of joy, Germany's Bundesbank warned Greece it needed reform to tackle its economic problems and the euro fell nearly half a U.S. cent.
Tsipras has promised to keep Greece in the euro and has toned down some of his rhetoric but his arrival in power would mark the biggest challenge to the approach adopted to the crisis by euro zone governments.
"We are delighted," said 47-year-old teacher Efi Avgoustakou. "We hope our expectations will be fulfilled," she said. "On Monday in class, we're not allowed to comment and take sides but we will be smiling."
With Greece's bailout deal with the euro zone due to end on Feb. 28, 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.
Financial markets have been worried a Syriza victory will trigger a new financial crisis in Greece, but the repercussions for the euro zone are expected to be far smaller than feared the last time Greeks went to the polls in 2012.
If Syriza ends up short of an absolute majority, Tsipras will have to try to form a coalition with smaller parties or reach an agreement that would allow it to form a minority government with ad-hoc support from others in parliament.
Negotiations are likely to begin immediately and both Panos Kammenos, the leader of the small Independent Greeks party and Stavros Theodorakis, head of the centrist To Potami party, said they would be willing to support an anti-bailout government.
If Syriza requires support to govern, it may find itself hostage to its partners' demands, raising questions over how durable a Tsipras government would prove.
STANDOFF WITH BERLIN
Tsipras has promised to renegotiate agreements with the European Commission, European Central Bank and International Monetary Fund "troika" and write off much of Greece's 320 billion-euro debt, which at more than 175 percent of gross domestic product, is the world's second highest after Japan.
Coming after the ECB's move to pump billions into the bloc's flagging economy, Sunday's result will stir consternation in Berlin. A senior lawmaker in Merkel's conservative party said the result showed Greek voters had turned away from austerity but he said Europe could not accept rejection of the bailout.
"We must not reward the breaching of agreements," Wolfgang Bosbach told the daily Osnabruecker Zeitung newspaper. "That would send completely the wrong signal to other crisis-stricken countries that would then expect the same treatment."
As well as renegotiating a debt agreement, Tsipras wants to roll back many of the measures demanded by the "troika", raising the minimum wage, lowering power prices for poor families, cutting property taxes and reversing pension and public sector pay cuts.
Markets had been jittery in the run-up to the vote but growing confidence that a deal could be reached has helped ease fears of a return to crisis.
U.S. investment bank J.P. Morgan said the result could weigh on markets but that it considered speculation over a possible Greek exit from the euro was "a stretch" and a negotiated deal appeared the most likely outcome.
It added: "our base case remains that a Syriza government or Syriza-dominated coalition would alter its platform to retain troika financing."
Syriza officials have said they would seek a six-month "truce" putting the bailout programme on hold while talks with creditors begin.
Greece, unable to tap the markets because of sky-high borrowing costs, has enough cash to meet its immediate funding needs for the next couple of months but it faces around 10 billion euros of debt repayments over the summer.