The election also saw a surge in support for Marine Le Pen's far-right National Front, which wants to ditch the euro and battle against what she calls the "Islamisation" of France.
The Socialists and their Green allies won around 46 percent of the vote, ahead of the 34 percent of ex-president Nicolas Sarkozy's right-wing UMP party and its allies, the final results released by the interior ministry showed.
Pollsters TNS Sofres, Ipos and OpinonWay agreed that the Hollande camp might win as few as 283 seats in the 577-seat National Assembly or potentially as many as 347. But potential allies in the anti-capitalist Left Front would take 13-20 seats and ensure a majority.
Hollande defeated Sarkozy in last month's presidential election and wants voters to give him a strong mandate to enact reforms as France battles Europe's crippling debt crisis, rising joblessness and a stagnant economy.
If next week's second round confirms Sunday's results, it will boost his status in Europe as champion of the movement away from German-led fixation on austerity towards growth, which he favours as the solution to the economic crisis.
Hollande's Prime Minister Jean-Marc Ayrault called for a "large, solid and coherent majority" for the Socialist party and its allies in the second round.
"Change is going to be around for a while," he said, echoing the Socialists' presidential election slogan.
The National Front won 13.6 percent of the votes, far above the 4.0 percent it achieved in the last parliamentary election in 2007.
But under France's first-past-the-post system, that would at best give it only three parliamentary seats and possibly none at all.
The Communist-backed Left Front, headed by firebrand anti-capitalist Jean-Luc Melenchon, won 6.9 percent of the votes.
Nationwide turnout was a modest 57 percent.
But the night marked a personal defeat for Melenchon who took Le Pen head-on in a bitter battle in a rundown former mining constituency near the northern city of Lille.
"It's normal to be disappointed but we must not be defeated," Melenchon said as he bowed out, while Le Pen claimed her victory meant her party was now France's third political power.
"Given the abstention rate and a profoundly anti-democratic electoral system that has for 25 years deprived millions of voters of MPs, we confirm our position tonight as France's third political force," Le Pen said.
Her niece Marion Marechal-Le Pen, also the granddaughter of party founder Jean-Marie Le Pen, beat left and right candidates in the southern Vaucluse district with 35 percent of the votes.
Although the party has not won a seat in parliament since 1986, Le Pen is seeking to build on her strong showing in the presidential vote and cement her party's place in national politics.
Melenchon won 11 percent of votes in the April-May presidential vote that was won by Hollande, while Le Pen won almost 18 percent of votes.
Centrist leader Francois Bayrou meanwhile appeared set to lose his seat in the southwest after his left and right-wing rivals beat him in the first round.
The mainstream left's strong score in the parliamentary vote means Hollande will likely not have to rely on support from the Left Front to make good on his electoral promises.
He has pledged to hire an extra 60,000 new teachers and to hit top earners with a 50-percent tax rate on some of their income.
The Socialists took control of the upper house of parliament, the Senate, last year.
Ayrault's interim government has taken a series of popular steps in the wake of Hollande's presidential victory in the May 6 run-off.
He has cut ministers' salaries by 30 percent, vowed to reduce executive pay at state-owned companies and lowered the retirement age from 62 to 60 for some workers.
But the UMP has hit back with warnings that the Socialists are preparing huge tax hikes to pay for what the right says is a fiscally irresponsible spending programme for Europe's second-biggest economy.
UMP leader Jean-Francois Cope said the Socialists were readying "the biggest-ever tightening of the screws on the middle class".
The economic backdrop is bleak for whoever wins the parliamentary vote: unemployment is at 10 percent, growth has stalled and the eurozone crisis has lurched back into the foreground.
More than 6,500 candidates were competing in Sunday's vote, which takes place over two rounds under a constituency-based simple majority system.
If no candidate wins more than 50 percent in the first round, any contender who scores more than 12.5 percent of the vote stays in the race for the second round.