Marine's Le Pen's far-right National Front fell flat with 13 percent of the vote.
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.
French far-left leader Jean-Luc Melenchon has decried the low participation rate in the first round of parliamentary elections, saying it shows France's volatile political situation.
Macron has banned all REM candidates from employing family members if elected and they must not perform consultancy work while lawmakers.
He continued: "Next Sunday, the National Assembly will embody the new face of our republic: a strong republic, a unified republic, a republic that listens to everyone, the French Republic".
Election officials on June 12 said that with almost all ballots counted, Macron's Republique en Marche party (Republic on the Move) had 28 percent of the vote in an election marked by a historically low turnout.
She said her party could win several seats in next week's second-round poll and urged voters to turn out in massive numbers.
It is all liable to bring a rush of blood to the head, and the greatest danger right now for Macron and En Marche is hubris.
The candidates of Republic on the Move include many newcomers in politics, like a retired bullfighter, a fighter pilot and a mathematical genius.
Macron, a former investment banker, wants what supporters describe as as a "big bang" of economic and social reforms, including an easing of stringent labour laws and reform of an unwieldy pension system.
NDA Presidential candidate Ram Nath Kovind meets Fmr PM Atal Behari Vajpayee
We haven't yet decided our stand on Kovind. "We will go by whatever is decided in the meet tomorrow", Prasad said. Apart from being a top lawyer, he's a two-time member of the Upper House of Parliament ( Rajya Sabha ).
"France is back", Edouard Philippe, the prime minister for Macron, said after the strong vote for the president's party, though he lamented the relatively light turnout, about 49 percent of the voting public, according to the Interior Ministry.
Socialist Party leader Jean-Christophe Cambadelis, who was eliminated in Paris, called the results "an unprecedented step back for the left as a whole and the PS (Socialist Party) in particular".
Benoit Hamon, the party's presidential candidate, was also knocked out of the running for his seat.
Under Marine Le Pen, the FN has consistently improved its electoral scores, notching up records in past regional, European Parliament and presidential elections.
The right-wing conservative party Le Republicains ("The Republicans") won about 21% of the votes.
The success of those parties, however, is likely to pale into insignificance should La Republique En Marche secure the mandate Macron requires to govern successfully.
"Today fewer than half of French people expressed a preference", he said. "They are neither forgotten nor wiped away".
The far-right formation had been hoping to pick up at least 15 MP seats so that it can be allowed to form a parliamentary group of their own, and is now projected to win only somewhere from one to five seats in the French National Assembly.
Emmanuel Macron is, once again, giving Theresa May a lesson in politics as the French president looks set to take an overwhelming majority in parliament.
Only six months ago the center-right Les Républicains Party was confidently angling for both the presidency and a governing majority in the legislature, and yet it obtained only 20 percent of the vote share and is now projected to obtain a meager 90 to 130 seats in the National Assembly-even less than it currently holds.
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