Europe's worst refugee crisis since the 1990s has strained the European Union's asylum system to the breaking point.
The poll showed Syriza set to win 25 percent of votes, just behind New Democracy on 25.3 percent. More than one in 10 voters remained undecided, meaning the final outcome is far from certain.
Vassiliki Thanou, an anti-austerity advocate who has argued against wage cuts for judges and court officials, will be sworn in as the country's first female prime minister.
Syriza says it is aiming for an outright majority, although the strength of its support is unclear due to a lack of surveys by leading pollsters in the past month.
The prime minister hopes to strengthen his hold on power in snap elections after seven months in office in which he fought Greece's creditors for a better bailout deal but had to cave in.
Tsipras is hoping to quell a rebellion in his leftist Syriza party and seal support to implement a tough bailout program.
The test ballot - a non-binding vote - showed 56 'no' votes and 4 abstentions on Tuesday.
"Finally, we have white smoke," a finance ministry official said.
A government spokeswoman said drafting of the bailout accord, which requires approval from Greece's fractious parliament, would start on Wednesday.