Tipped to win, the opposition Social Democrats scored 17.7 percent, while their eurosceptic Finns Party rivals were at 17.5 percent, according to almost complete results published by the justice ministry.
After losing some ground when hardliner Jussi Halla-aho took the party's reigns in 2017, the Finns have made rapid gains in recent months when a number of cases of sexual abuse of minors by foreign men stirred anti-immigration sentiments.
With the Social Democrats, Green League and Left Alliance holding 76 of the 200 seats in the Parliament, the Social Democrats will likely have to open the door to either the National Coalition or the Centre Party.
The Social Democrats' Antti Rinne has previously said his party would find it "very difficult" to enter a coalition with the Finns Party. "SDP is the prime minister's party", he told supporters in central Helsinki, Reuters reported.
The left-wing party leads Finland's two main opinion polls with about 19 percent of the vote, having campaigned against the steep cost-cutting of Centre Party Prime Minister Juha Sipila and his Finance Minister Petteri Orpo - leader of the conservative National Coalition Party.
The Social Democratic Party had been leading in opinion polls ahead of the election, at around 19%.
Rinne on Sunday announced he will send all other parties a list of questions, including ones about values, to decide which parties to invite to the coalition formation talks. The Finns Party focused its campaign on immigration, urging people to "Vote for some borders", and on climate change, where it denounced the "climate hysteria" of other parties and pledged that citizens should not have to pay for efforts to contain global warming.
"The Finns Party has had an effect on the way that we speak about immigration", political commentator Sini Korpinen said.
Rinne has been a staunch opponent of the National Coalition's austerity policies over the past four years.
However, in a tacit acknowledgement that the public mood has turned against further belt-tightening, Orpo has insisted the economy is now strong enough to allow for some more generous public spending.
As required by Finnish law, election officials asked the first voter to confirm the ballot box was empty before it was locked and voting could begin.
At stake in Finland is the future shape of the country's welfare system, a pillar of the social model across the Nordics, which the leftists want to preserve through tax hikes and the center-right wants to see streamlined because of rising costs.
Underscoring a growing confidence among the far-right in Europe, anti-immigration parties, including the Finns, have announced plans to join forces after the May 26 European Union election in a move that could give them a major say in how the continent is run.
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