At the start of the ceremony former prime minister Laurent Fabius greeted Macron as "a man of his time", adding that he now faced the task of leading France.
Europe and the world need a "strong France", Macron said.
"We need Europe and it will be reformed and relaunched because it protects us and allows us to project our values in the world," he went on.
Macron promised to restore confidence to the French people, to convince them that "our country ... has all the resources to be in the first rank of nations".
Swearing he would not concede on any of his election commitments, he promised to support business, innovation and creativity without forgetting those who feel left behind by a changing world.
He pledged to renew France's institutions and warned against "taking refuge" behind outdated attitudes and habits.
Tribute to predecessors, especially Hollande
The new president paid tribute to his predecessors, starting with General Charles De Gaulle and ending with François Hollande, whose work for the Paris climate change deal he hailed.
Hollande, who had left the Elysée earlier to go to the headquarters of his Socialist Party, told supporters there to greet him that he left France in a much better state than it was in when he took office.
He listed the challenges before France today, including the growth in migration, climate change and the "excesses of global capitalism".
"We are not fighting just for this generation but for those to come," he declared. "We must build the world our youth deserves."
"I know the French expect much of me," Macron said, finishing his speech "As for me, I will get to work; Vive la France! Vive la République!"
As music by Mozart, Offenbach and Brahms played, the new president toured the room shaking the hands of the 300 people who attended the ceremony.
They included members of his Republic on the Move! party, civil servants and polticians, the partners of a Socialist MP who died at his last election rally and a police officer murdered during the campaign, and business representatives, including Pierre Gattaz, the head of bosses' union, Medef.
After the swearing-in a 21-gun salute was fired from the Invalides military hospital on the other side of the River Seine, as tradition dictates, and the new president reviewed the troops at the Elysée to the strains of the Marseillaise.
As the heavens opened, Macron left for the Arc de Triomphe on Paris's Champs Elysées where he laid a wreath and relit the flame at the tomb of the unknown soldier.
After the ceremony he greeted military veterans before shaking the hands of people in the crowd around the famous landmark, then driving down the Champs Elysées, waving to onlookers.
During the afternoon he was to visit seriously wounded soldiers at a military hospital near Paris before meeting Paris mayor Anne Hidalgo at the capital's city hall at 5.00pm.