Two of the children and one adult were in hospital in a life-threatening condition while the other victims were less seriously hurt.
A video of the attack, taken by a bystander and verified by Reuters, showed the assailant jump a low wall into a children's playground and repeatedly lunge at a child in a stroller, pushing aside a woman who tries to fend him off.
Prime Minister Elisabeth Borne said the suspected attacker, who is in police custody, was a 31-year old Syrian citizen who was granted asylum in Sweden 10 years ago.
He had entered France legally, she said, and was carrying Swedish identity documents and a Swedish driving licence.
The alleged culprit defined himself as a Christian when requesting asylum in Sweden and France, according to police quoted by French media.
Identified as Abdalmash H, the alleged attacker has had refugee status since 2013 in Sweden, where he married a Swedish citizen with whom he shares a three-year-old son, according to the reports.
The young man had been living in France since late 2022 and has no police records.
At the time of his arrest, Abdalmash H was reportedly carrying a cross and a Christian prayer book.
The local prosecutor leading the investigation said there was no indication that terrorism was the assailant's motivation.
He was under investigation for attempted murder.
The four children were just toddlers, aged between 22 months and three years, Annecy prosecutor Line Bonnet-Mathis told reporters.
One of them was a British citizen and another was Dutch, Bonnet-Mathis said.
As the assailant, who wore a blue-chequered headscarf and sunglasses, slashed at his victims, one bystander tried to stop him by throwing his backpack at him, the video showed.
The incident took place about 9.45am on Thursday in Le Paquier park in Annecy, a town in the French Alps.
"The nation is in shock," President Emmanuel Macron said on Twitter, calling the attack "an act of absolute cowardice".
"He clearly targeted the babies," a witness who gave his name as Ferdinand told BFM TV.
"Mothers were crying, everybody was running," said George, owner of a nearby restaurant.
Another video, also verified by Reuters, showed the assailant, who was shot at by police as they sought to detain him, being overpowered by officers.
He was not injured, the prosecutor said.
Several witnesses described Le Paquier park as an usually tranquil place popular with tourists for its stunning views of Lake Annecy and the mountains.
"It's a place where babysitters and parents take young children to play. I often see around 15 toddlers there in the morning, and the atmosphere is fantastic," said Yohan, who works at an ice-cream parlour just opposite the park.
France has been shocked by a number of violent incidents over the past few months, including the fatal stabbing last month of a nurse in the northern town of Reims.
Also last month, a drunk driver accidentally killed three policemen.
Macron has denounced what he calls a "de-civilisation process" in the country, while opposition lawmakers say his government has been too lax on law and order.
"Nothing more abominable than to attack children," National Assembly Speaker Yael Braun-Pivet said on Twitter.
Parliament observed a minute of silence to mark the incident.
with EFE
