* League leader says party has right to govern
* Anti-establishment 5-Star also claims leadership role
* Former premier Renzi quits as head of center-left
* Italian assets under pressure (Adds Renzi quotes, market reaction)
ROME, March 5 (Reuters) - Two anti-establishment leaders made early plays to govern Italy on Monday, triggering concern in the euro zone following an inconclusive election where voters shunted mainstream parties to the sidelines.
With the euro zone's third-largest economy seemingly facing prolonged political instability, the anti-immigrant League claimed the right to rule after its center-right alliance won the largest bloc of votes.
"We have the right and duty to govern," League leader Matteo Salvini told a news conference.
Investors should have no fear, he said, but the prospect of a eurosceptic-led administration promising to ramp up spending hit shares, bonds and the euro.
Luigi Di Maio, the head of the biggest single party, the anti-establishment 5-Star Movement, said his group had emerged as the clear winner on Sunday, capturing around a third of the vote, and should take the helm of the next government.
"We're open to talk to all the political forces," 31-year-old Luigi Di Maio said in a statement. "We feel the responsibility to give Italy a government (as) ... a political force that represents the entire nation."
With the vote count almost complete, none of the three main factions had enough seats to govern alone. President Sergio Mattarella is expected to open formal coalition talks in April, with early elections possible if no accord is found.
The ruling center-left Democratic Party (PD) suffered a crushing defeat, prompting former prime minister Matteo Renzi to resign as party leader. But he said the PD would play no part in the next administration.
"The Italian people have asked us to be in opposition and that is where we will go," he told reporters. "We will never form a government with anti-system forces," he added, referring to the 5-Star and the far-right League.
The rightist alliance that also includes former prime minister Silvio Berlusconi's Forza Italia (Go Italy!) was on course for about 37 percent of the vote -- but for the first time the League emerged as the senior partner.
The role reversal marks a bitter personal defeat for the billionaire media magnate and his party, which took more moderate positions on the euro and immigration while the League campaigned on a fiercely anti-migrant ticket.
Berlusconi, who has not been seen in public since the vote, met Salvini on Monday in his villa near Milan.
Afterwards, Forza Italia said it was open to welcoming in other parties to create a center-right government, but it was not clear where such support might come from, with the bloc seen some 49 seats short of a majority in the 630-seat lower house.
Before the encounter, Salvini told reporters that, while not interested in a broad "minestrone" coalition, the League would be willing to talk to all parties. The party's economics chief Claudio Borghi said their first contact would be with 5-Star.
When the PD took office in 2013, Italy was deep in recession. The party subsequently oversaw a modest recovery, but many Italians did not see any improvement in their lot, with the economy still some 6 percent smaller than it was a decade ago and unemployment stuck near 11 percent.
The PD also paid the price for widespread anger over an influx of more than 600,000 migrants over the past four years.
The European Union has insisted that Italy take charge of the newcomers, letting hardly any of them move to other EU states. This stance has helped erode Italian faith in Europe and French President Emmanuel Macron said on Monday the election result was a reminder of problems caused by migration.
"Italy has, it's undeniable, suffered for months and months under the pressure of migration," Macron said in Paris.
Prolonged political stalemate could make heavily indebted Italy the focus of market concern in Europe. Investors dumped Italian government bonds on Monday and the euro was under pressure, while the Milan bourse dipped 0.5 percent.
"It's not a positive result because we have seen the ascent of parties that create some concern in terms of reforms, their views on the euro and their views on forming alliances," said Maria Paola Toschi, global market strategist at JP Morgan Asset Management.
Formed in 2009, 5-Star has fed off public fury over institutional corruption and economic hardship. Its flagship proposal in the election campaign was a minimum monthly income of up to 780 euros ($963) for the poor, helping it draw overwhelming support in the underdeveloped south.
By contrast, the right dominated in the wealthier north, with support for the center-left largely concentrated into a narrow stretch of central Italy, leaving the country more politically divided than at any time in recent history.
"It's what we expected, political deadlock. A government is needed," Rome resident Giuseppe Bruni told Reuters. "We are in the hands of the president and the willingness of political parties to put a government together." ($1 = 0.8097 euros)
(Additional reporting by Gavin Jones, Francesca Piscioneri, Isla Binnie and Philip Pullella Editing by Catherine Evans)