La Republique en Marche (The Republic on the Move, LREM) and allies, with an estimated 32.9 per cent of the vote, is expected to take between 400 and 440 of the assembly's 577 seats in next week's second round, according to a Kantar Public-onepoint forecast based on partial results.
The conservative Republicans had 16 percent, followed by the far-right National Front with 14 percent.
In the last legislative elections in 2012, turnout in the first round stood at 57 percent.
The result was also disappointing for radical leftist Jean-Luc Melenchon, who had hoped for a parliamentary breakthrough on the back of his 19.6 percent vote in the presidential election.
Rivals began sounding the alarm regarding the majority, with Jean-Christophe Cambadelis, head of the Socialist Party, remarking that opposition almost no longer exists, and Marine Le Pen calling the abstention rate "catastrophic".
Prime Minister Edouard Phillipe, who was appointed by Mr Macron last month, announced "France is back" as the results rolled in.
The 39-year-old's La Republique En Marche (LREM) party, is hoping to make huge gains and inflict a further blow on the country's traditional ruling parties. Opponents occupying the remaining seats would represent a fragmented opposition, most without the 15 seats needed to even get speaking time, funding or other ways to weigh on policy.
She also slammed the electoral system as unfavourable to smaller parties like hers.
The head of the conservative Republicans party, Francois Baroin, also urged voters to turn out in larger numbers next week to help ensure that Macron's party faces a robust opposition.
Such a share would give Macron one of the biggest parliamentary majorities for 60 years.
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Last year's finalist could bounce back immediately after suffering two breaks in the third set, reports Xinhua news agency. The eighth-seed shook off a disastrous first set to beat Spain's Fernando Verdasco 0-6, 6-4, 6-4, 6-0 on Monday.
German Chancellor Angela Merkel's office tweeted her congratulations to Macron for his party's victory.
Mr Macron's party was established just over a year ago and many candidates have little or no political experience.
France's right wing Republican party, allied with the Union of Democrats and Independents, secured 18.80 percent.
The opposition and French press expressed concern over what the left-wing Liberation daily called the "quasi-Stalinist result". But they've got their work cut out: all signs suggest the French President will win convincingly next Sunday.
Pollsters project Macron's alliance could win as many as three-quarters of the seats in the lower house after next week's second round of voting.
Sunday's projections pointed to another torrid night for the two main traditional parties, which have suffered high-profile defections to Macron's government and party.
Macron wants a powerful mandate to push through plans to reduce worker protections to boost hiring, boost security and clean up corruption in politics. Less than 50 percent of the 47.5 million electors cast ballots - showing that Macron has limited appeal to many voters.
Seven to 12 seats will be claimed by other candidates.
His party fielded political newcomers in around 200 constituencies, some of whom felled heavyweights of the left and right in the first round.
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