Socialist President Francois Hollande looks set to consolidate his position with a left-wing majority in parliament after a first-round vote today.
He may be able to govern without relying on hard leftists opposed to closer European integration.
Initial projections by polling agencies based on a partial vote count suggested his core Socialist bloc could win 283 to 329 seats in the 577-member National Assembly in next Sunday's runoff, shifting the lower house to the left for the first time in a decade.
With Greens allies, the left would have 295 to 347 seats, the CSA polling institute forecast, well ahead of the mainstream conservatives with 210 to 263 seats.
Mr Hollande needs 289 seats to have an outright majority.
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.
The vote was another symbolic advance for the left after it took control of the Senate in 2011 and won the presidency in May for the first time since 1988.
A low turnout, less than 60%, which analysts say plays against the left, could still curb what might have been a bigger triumph for Mr Hollande as he seeks to steer France through the eurozone's debt crisis.
"There is a winner on the left, but not a big winner. It is a relative majority and not an absolute majority. In a crisis there was a need for an absolute majority," said analyst Stephane Rozes at the CAP political consultancy.
The parliamentary balance is hard to predict accurately because any candidate winning more than 12.5% of registered voters can go through to the 17 June second round, meaning three candidates may face off in many constituencies.
The far-right National Front won about 13.4% of the popular vote, less than its leader Marine Le Pen's 17.9% presidential score, and is projected to have at most three deputies.
The anti-immigration party will be present in fewer runoffs than it had hoped due to the low turnout.
In one of today's most high-profile battles, Ms Le Pen bested hard leftist Jean-Luc Melenchon in Henin-Beaumont in the northern Calais region, and a Socialist candidate may pip the Left Party leader for second place on the ballot for round two.
In another closely watched race, Mr Hollande's former partner Segolene Royal, who ran for president in 2007, was narrowly ahead of a maverick leftist rival in the western seaside town of La Rochelle and faces a difficult three-way runoff.