With over 70% of votes counted, Mr Tsipras held a lead of 35.46%, to the conservative New Democracy Party's 28.27%.
Counting is continuing, but the party is predicted to fall just short of securing an overall majority.
The far-right Golden Dawn is expected to come in third, with seven per cent.
Mr Tsipras says his Syriza party will form a coalition with the small nationalist Independent Greeks (ANEL) party within days.
Even as the votes were still being counted, the leader of the Conservative New Democracy party, Vangelis Meimarakis conceded defeat:
"The race for the first party has now been completed, with great dignity and in a civilised fashion. The final result is still evolving. It would appear that Syriza and Alexis Tsipras do have the first place. I do congratulate him."
In an impassioned speech claiming victory, Alexis Tsipras told crowds the result appear to justify the difficult decisions he had taken.
"The Greek people gave a clear mandate so that we can put an end to all that is keeping us tied down to a yesterday, so that we can continue with our beautiful struggle, a struggle that we started seven months ago."
The snap election was called after Syriza lost its majority in August following the signing of a new financial bailout deal with international creditors.
Alexis Tsipras's popularity plummeted when he agreed to a deal with European leaders implementing more austerity measures - in contrast to his anti-austerity campaigning ahead of the last election.
Former Greek finance minister, Yanis Varoufakis quit Syriza after July's bailout referendum, having been earlier pulled by Mr Tsipras from the country's bailout negotiating team.
Before poll results emerged, he launched a scathing attack against his former party.
"I'm sorry but Syriza doesn't exist any more. What we've had after the 12th July is a surrender of our government, which followed with the deconstruction of Syriza."
In interviews leading up to the election, Mr Tsipras said he had put his country above his party.
He said if he had not agreed to the three-year bailout, Greece would probably have had to leave the eurozone.
It's the country's fifth election in six years, and voter turnout was at historic lows of about 54 per cent.
Many Greeks, like this voter, are calling for more unity in the government.
"We Greeks must be united and the parties must cooperate so that there is finally a positive outcome. As Greeks we must be united and together fight those who are after us."
The new government will be under pressure to deal with the migrant and refugee crisis which had threatened to overshadow the election.
As large numbers arrived on the small tourist island of Leros this week, the local Mayor had threatened to stop voters taking part in the general election unless he received more support from Greek government.
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