By Stelios Bouras And Alkman Granitsas 

ATHENS--Greek leftist party Syriza is set to win Sunday's election but may not have enough support to form a government on its own, according to an official projection from the interior ministry.

The antiausterity party is projected to win 150 seats in Greece's 300-seat parliament with the backing of 36.5% of voters, the ministry said, adding that it won't be known whether it will have an outright majority--or fall just short--until the all the votes are counted in the early hours of Monday.

Conservatives New Democracy are projected to win 76 seats after being backed by 27.7% of the electorate, the ministry said, with ultranationalists Golden Dawn coming in third place with 17 seats.

A total of seven parties are projected to surpass the 3% threshold needed to enter parliament, the ministry added.

Greeks headed to the polling booths Sunday for the third time in less than three years in a fraught and historic national election that could determine the country's future in the eurozone and reverberate across Europe.

The result of the elections is expected to have lasting repercussions for both Greece and the eurozone. Syriza, which emerged as the main opposition party in mid-2012 at the depths of the nation's debt crisis, has promised to tear up the austerity program that Athens pledged in exchange for a EUR240 billion ($269 billion) bailout from international creditors.

Since first seeking a bailout in 2010, Greece has undertaken a broad sweep of economic overhauls and cutbacks that have helped mend its public finances and nudged the economy back to growth following six years of deep recession. Those cutbacks have come at a cost: Some 25% of Greeks remain jobless, while a quarter of households live close to the poverty line.

Syriza has promised to change all of that, pledging immediate relief to the poor, rolling back unpopular taxes and negotiating a debt write-down with the country's creditors to free up spending on social programs.

"Today the Greek people are being called upon to take the final decisive step needed: a step toward restoring hope, to ending fear and to restoring democracy and dignity to our country," Syriza leader Alexis Tsipras told journalists earlier Sunday after casting his vote in a working class district of Athens.

However, Syriza's promises to roll back many of those overhauls and its demands for debt relief from Greece's international creditors have stoked fears of an open clash with eurozone partners that could lead to the country's exit from the common currency and once again rattle financial markets world-wide. In the past three months, Greece's main stock index has shed roughly a fifth of its value, while nervous depositors have withdrawn several billions of euros from the country's banking system.

A Syriza victory would also be closely watched by other antiausterity parties in Europe--on the left and the right--that have been gaining ground in the past year. In Europe-wide parliamentary elections last spring, voters fed up with years of cutbacks, rising unemployment and a shrinking social state, strongly backed new and fringe antiestablishment parties such as France's National Front and Spain's newly created Podemos party in a reaction to Europe's old guard.

In Greece, those feelings run particularly high. "Europe is self-destructing," said Polyxeni Konstantinou, a 56-year-old public-sector worker voting in central Athens. "I voted for Syriza because I hope that it will help change the tragic circumstances that now govern Europe. Will Syriza be able to achieve everything it says? Probably not. But whatever it does achieve, then that will be good for Europe."

Under Greece's electoral law, the party that wins the most votes is awarded 50 bonus seats in the country's 300-member Parliament, a measure aimed at facilitating the stability of an elected government. The 50-seat bonus, which Syriza appeared set to win, could put the leftist party within reach of winning a majority in parliament, allowing it to forgo coalition partners.

If Syriza does fall just shy of a majority, though, it would need to turn to another party to help govern, putting the third-place finisher in a position to become a kingmaker.

The favorite to take third is To Potami, a middle-of-the-road party launched by a prominent television journalist in March that has rapidly drawn voters disillusioned with Greece's traditional political parties. It is running neck-and-neck for third place with the ultranationalist Golden Dawn in the polls, at between 6% and 7%.

To Potami's leader has been coy about whether he would agree to form a coalition with Syriza, saying he would prefer some kind of national unity government of the top parties.

Stelios Bouras contributed to this article.

Write to Nektaria Stamouli at nektaria.stamouli@wsj.com