The True Finns finished just behind the conservative NCP and the Social Democrats on around 19%.
While the Social Democrats have called for changes on EU bail-outs, including the planned Portuguese rescue, True Finns opposes the plans altogether.
A hostile Finnish government could theoretically veto the package.
Unlike other eurozone countries, Finland's parliament can vote on whether to approve the measures.
Correspondents say the increased sway of Euro-sceptics in Finland's parliament could hold up any further bail-out deals.
As the biggest party, the NCP is tipped to lead the next government with former Finance Minister Jyrki Katainen likely to become prime minister of whatever coalition emerges, replacing Mari Kiviniemi of the Centre Party.'Invitation to talks'
The anti-immigration True Finns won 39 seats in the 200-member parliament, final results showed.The conservatives' Jyrki Katainen is tipped to be the next prime minister
That put it five seats behind the conservative National Coalition Party (NCP) - part of the current centre-right government and a strong advocate for European integration - and just three behind the opposition Social Democrats.
The Centre Party - previously the largest party in parliament - won just 35 seats, down 16 from the last election in 2007.
The strong showing for the True Finns meant the anti-euro party would at least "get an invitation to talks" on a new government, Reuters quoted party leader Timo Soini as saying.
Celebrating the NCP's success, Jyrki Katainen played down suggestions that Finland would now cause difficulties for the eurozone.
"Finland has always been a responsible problem solver, not causing problems," he said.
"This is about a common European cause. After the elections, the biggest parties will begin to look for common ground."EU 'squanderers'
Tampere University political analyst Ilkka Ruostetsaari told AFP news agency the election outcome was astonishing."The True Finns' victory, surpassing every poll and every expectation of a drop on election day... plus the total collapse of the Centre - the whole thing is historic," he said.
Opinion polls had predicted a strong result for the True Finns but were giving the party around 15%, not 19%. In the 2007 election, the party had won just 4%.
With its charismatic leader Mr Soini, the party rejects rescue funds for EU "squanderers", as well as opposing immigration.
Speaking on Finnish TV, the True Finns' leader said he wanted to change the terms of the bail-out for Portugal.
"The package that is there, I do not believe it will remain," he said.
At the same time, he sought to assure other EU states that his party posed no threat.
"We are not extremists, so you can sleep safely," he said.
Analysts attribute much of his party's success to disenchantment with the big three mainstream parties who have run Finland for decades.
"It's a multiple protest," journalist Timo Harakka of Finnish TV YLE told BBC World Service.
"It's a no confidence vote for how things have been run so far. Concerning Europe for instance, or the economy.
"It seems The True Finns' just had a magical moment right now," he said, adding that, "negotiations for the government will be really strenuous for the coming months."
Some Finns expressed concern about the surge in support for the True Finns.
"They have strict opinions about everything," one young woman voter, who gave her name as Eevi, told Reuters.
"Finnish people have always been very open, I wonder why we are now pulling off, closing up again."
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