Projections show La Republique en Marche (Republic on the Move) and its MoDem ally set to win up to 445 seats in the 577-seat National Assembly.
The final outcome will be decided at a run-off next Sunday.
Mr Macron’s party was established just over a year ago and many candidates have little or no political experience.
With all the ballots counted, Mr Macron’s LREM and MoDem had won 32.3% of the vote.
The centre-right Republicans had just under 16%, while the Socialists, previously France’s ruling party, had won just 7.4%.
The far-right National Front (FN) had 13.2%, followed by the far-left France Unbowed on just over 11%.
But turnout was sharply down, at 48.7% compared with 57.2% in the first round in 2012, which analysts say reflected a sense of resignation among Mr Macron’s opponents.
Mr Macron, 39, defeated FN leader Marine Le Pen in the presidential run-off last month.
He needs a majority to push through the changes that he promised in his campaign.
He has already left an impression around the world, in particular for standing up to US leader Donald Trump on issues like climate change.
After the projections were announced, a government spokesman said voters had shown they wanted to move fast on major reforms.
Marine Le Pen blamed her party’s poor performance on the low turnout, saying France’s electoral system, which favours larger parties, needed to be reformed.
“This catastrophic abstention rate should raise the question of the voting rules which keep millions of our compatriots away from the polling stations,” she said.