Antti Rinne’s left-wing party had 17.8 per cent of the vote after more than 97 per cent of ballots were counted on Sunday night.
The nationalist Finns Party were in second place, with 17.6 per cent.
“For the first time since 1999 we are the largest party in Finland ... SDP is the prime minister party,” Mr Rinne said.
If the former union leader becomes prime minister, he will be Finland’s first left-wing leader in 20 years.
He expressed some disappointment over the SDP’s performance during the extremely tight count, where the parties were almost neck-a-neck.
“I have to make a honest confession: I hoped still for a better result,” he said.
The preliminary tally would hand 40 seats in the country’s 200-member legislature to the SDP, while the Finns Party will have 39.
The conservative National Coalition Party won 37 seats and the liberal Centre Party has 31.
Juha Sipila, the former prime minister and Centre Party leader, resigned last month, as his entire centre-right coalition government collapsed.
The politician left the post after it became clear he could not secure a healthcare reform package which formed part of his plan to cut government spending.
In contrast, Mr Rinne has promised to raise taxes to fund welfare and tackle economic inequality.
The nationalist Finns Party has argued that the government spends too much on immigration and on fighting climate change.
Sunday’s vote, held six weeks ahead of elections for the European Union’s (EU) legislature, was being watched in other European capitals for indications of how a broad bloc of eurosceptic parties might do across the EU.
Additional reporting by agencies
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