* Hollande set for lower house majority, may not need far left
* Socialists and non-Communist allies projected to win
* Hard left to win far more seats than extreme right
PARIS, June 10 (Reuters) - French President Francois Hollande was on track for a Socialist-led majority in parliament after a solid win in a first-round vote on Sunday that should free him from having to rely on hard leftists hostile to European integration.
Hollande's Socialist bloc looks likely to win the 289 seats needed for an outright majority in the 577-seat National Assembly in next Sunday's run-off, and almost certain to do so with its Greens Party allies, polling institutes said.
While conservatives said this was no "pink wave", winning power in the lower house for the first time in a decade would be a triumph for the left a year after it won control of the Senate and weeks after recapturing the presidency after 17 years.
Socialist Prime Minister Jean-Marc Ayrault warned against over-confidence, telling voters: "This is just the first round. Everything hinges on next Sunday. Change is beginning."
Hollande, who won the presidency five weeks ago due to a rejection of conservative Nicolas Sarkozy and his failure to curb 10 percent unemployment, asked for a coherent majority as he steers France through the resurgent euro zone debt crisis.
The government of Europe's second-biggest economy is preparing budget adjustments, including possible spending cuts to take account of sickly growth, as well as taxes on the wealthy in a broad reform in the weeks ahead.
In the longer term he is under pressure from Berlin to give European Union institutions more control over national budgets and move towards a fiscal union - measures that Communists and other radical leftists would oppose in parliament.
Conservative lawmakers will probably vote against tax increases, though they could back legislation to ratify a European budget discipline pact that the far left opposes.
Hollande can count on the strongly pro-European Greens to back most of his legislation.
TURNOUT HEADED FOR RECORD LOW
Initial projections based on a partial vote count suggested Hollande's core Socialist bloc could win 283 to 329 seats in the 577-member National Assembly in next Sunday's runoff.
With the Greens, the government would have 295 to 347 seats, the CSA polling institute forecast, well ahead of the mainstream conservatives with 210 to 263 seats, and enough to give Hollande the free hand he wants to govern.
The Left Front of hard leftists and Communists could still make an impact with a projected 13-19 seats but its leader, Jean-Luc Melenchon failed to win a seat.
The Ipsos institute drew similar conclusions from its own projections based on a partial vote count, while TNS Sofres was the only pollster that put the Socialists and Greens just short of an absolute majority in a worst-case scenario.
A turnout of less than 60 percent curbed what might have been an even bigger victory for the left, limiting the number of constituencies where the far right was able to get through to the second round and split the right-wing vote.
Candidates need to win more than 12.5 percent of registered voters to go through to the runoff.
The far-right National Front won about 13.4 percent of the overall vote, less than its leader Marine Le Pen's 17.9 percent in the presidential first round, and is projected to have at most three deputies.
Le Pen stormed into first place with 42.3 percent in her constituency of Henin-Beaumont, a decaying working-class bastion in northern France, forcing Melenchon into third place. He dropped out in favour of the second-placed Socialist.
Another family member, Marion Marechal Le Pen, came first in a southern constituency aged just 22. "Tonight the French people have banged their fists on the table and shown that the National Front vote is one of conviction, not of protest," she said.
The National Front has stolen votes from Sarkozy's UMP party in recent months and the party, divided and largely leaderless since its May 6 defeat, was dealt a blow by Sunday's result.
"We must unite and mobilise to prevent the irreparable in the second round - by irreparable I mean giving the left all the power to implement its disastrous programme," UMP secretary-general Jean-Francois Cope said.
DRIZZLE COMPOUNDS GLOOM
Among Socialist candidates, Foreign Minister Laurent Fabius and European Affairs Minister Bernard Cazeneuve were re-elected, winning more than 50 percent in the first round.
Finance Minister Pierre Moscovici, who like Hollande is staunchly pro-European, was on track to win a runoff in a three-way contest, a relief for Hollande who has said that ministers who lose their parliamentary races must resign from government.
Moscovici will be one of the architects of the tax reform Hollande plans by the autumn to implement his tax-and-spend programme aiming to revive growth without imposing austerity.
Some voters remain sceptical of Hollande's programme.
"There are already so many taxes in France. It's hard enough to run your own business as it is and with the Socialists there will just be more taxes," Cambodian-born Sor Chin-Run said as he voted for a conservative candidate in the eastern Doubs region.
Turnout in legislative elections has dwindled since France synchronised the presidential and parliamentary terms a decade ago.
Hollande was seeking a show of strength as he lobbies Berlin for measures to promote economic growth to accompany a budget discipline treaty agreed by EU leaders earlier this year.
Merkel has made signing up for an eventual fiscal union in Europe her condition for agreeing to Hollande's pro-growth ideas and rebuffed calls by Paris and others for a banking union with a joint deposite guarantee and the issuance of common euro zone bonds.