With nearly all of the votes counted in Sunday`s poll, Syriza looks set to have 149 seats, just two short of an absolute majority.
Syriza`s victory has raised fears about Greece`s future in the euro.
The currency fell to $1.11 against the US dollar following the result - the lowest level in more than 11 years.
Greece has endured tough budget cuts in return for its 2010 bailout, worth €240bn (£179bn; $268bn) and negotiated with the so-called troika - the European Union, International Monetary Fund (IMF) and European Central Bank (ECB).
The economy has shrunk drastically since the 2008 global financial crisis, and increasing unemployment has thrown many Greeks into poverty.
Syriza`s election result will send shockwaves through Europe, the BBC`s Gavin Hewitt in Athens says.
A majority of voters in Greece have essentially rejected a core policy for dealing with the eurozone crisis as devised by Brussels and Germany, our correspondent adds.
The election result is expected to be one of the main issues at Monday`s meeting of 19 eurozone finance ministers.
In Germany, Bundesbank President Jens Weidmann said he hoped "the new Greek government will not make promises it cannot keep and the country cannot afford".
Belgian Finance Minister Johan Van Overtveld was quoted by VRT network as saying that Greece "must respect the rules of monetary union", although he added that there was room for some flexibility.
In Italy, EU Affairs Minister Sandro Gozi said: "After this vote we will have new opportunities to pursue change in Europe to create growth and investment and fight against unemployment."
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