By Stelios Bouras 

ATHENS--The head of Greece's leftist Syriza party, set to win Sunday's elections, promised Greeks to end the austerity that sent the country into an economic tailspin without causing any "catastrophic break" with Europe.

"Greek people have written history. The Greek people have sent a strong undisputable mandate for the country to leave behind it austerity," Alexis Tsipras told thousands of flag-waiving supporters in central Athens.

"The new government will negotiate with creditors a fair viable solution--a solution for Greece to end its vicious cycle."

According to an official projection from the Interior Ministry, Syriza came in first in the elections, but may not have enough support to form a government on its own.

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

The conservative New Democracy is projected to win 76 seats after being backed by 27.7% of the electorate, the ministry said, while the ultranationalist Golden Dawn party seem headed for third place, taking 17 seats.

A total of seven parties have managed to get above 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 tumultuous national election that could determine the country's future in the eurozone and reverberate across Europe. Years of austerity by Greece--demanded by international creditors in exchange for an economic rescue package worth 240 billion euros--resulted in a six-year recession and unemployment of more than 25%.

In a speech admitting defeat, Prime Minister Antonis Samaras said he is handing over a country in much better condition that it was when he took over more than two years ago. He has warned that Mr. Tsipras would put Greece's place in the eurozone at risk with his promises that the country could abandon austerity and demand from international creditors that more than half of the country's debt be reduced.

"The Greek people have spoken and we respect their decision. I hope that my forecasts do not come true but I was obligated to make them," said Mr. Samaras.

Write to Stelios Bouras at stelios.bouras@wsj.com