The attacker entered a tobacconist shop in Romans-sur-Isère, near Grenoble, and stabbed the owners and a customer.
He then attacked more people at two other shops before being arrested.
Prosecutors said the suspect was a Sudanese refugee in his 30s who lived in the town. Two other people have also been arrested, police said.
At the time of his arrest on Saturday, the man was "found on his knees on the pavement praying in Arabic", prosecutors said.
David Olivier Reverdy, of the National Police Alliance union, said the man had asked police to kill him.
Counter-terrorism prosecutors said they had launched an investigation into "murder linked to a terrorist enterprise".
The suspect was not previously known to the police or intelligence services, news website France Bleu reported.
On a visit to the town, Interior Minister Christophe Castaner said: "This morning, a man embarked on a terrorist journey."
The butcher's shop owner, Ludovic Breyton, said: "He took a knife, jumped over the counter, and stabbed a customer, then ran away. My wife tried to help the victim but in vain."
In a statement, the prosecutor's office said initial investigations suggest the attacker had "a determined murderous course aimed at seriously disturbing public order by intimidation or terror".
During a search of the suspect's home, "handwritten documents with religious connotations were found", it said.
Prosecutors said they arrested a second Sudanese man at the suspect's home and on Sunday revealed that a third person - "a young Sudanese man from the same household" - was also in custody.
Two of the wounded are said to be in a critical condition.
President Emmanuel Macron described the attack as an "odious" incident that further saddened a country already going through an ordeal.
"My thoughts are with the victims of the Romans-sur-Isère attack - the injured, their families," he tweeted.
Mr Macron promised that "light will be shed" on the crime.
France is currently in lockdown because of the coronavirus pandemic. People are only allowed out to buy basic necessities or for exercise.
The country has been on high alert since 2015, when Paris was hit by a series of attacks attributed to the Islamic State group.
