When Belgacem got to Orly Airport in the southern outskirts of Paris, he went to the South Terminal, carrying a weapon and a can of gas in his rucksack, Molins said.
Addressing reporters at the airport, Le Roux said the man had tried "but not succeeded" in his bid to grab the soldier's rifle who was on patrol with two male colleagues.
The soldier was threatened by the man but "struggled and managed to keep her weapon, fighting against her attacker on the ground", a statement posted to Facebook by the French military read. With the soldier down on her knees, the suspect rose enough for the other two soldiers to shoot him several times.
NPR's Eleanor Beardsley reports that French interior minister Bruno Le Roux said the man was the same person who had shot a police officer earlier in the day during a routine traffic stop, and who had then hijacked a auto.
Authorities say Belgacem then headed to a bar in Vitry-sur-Seine, where he had spent the night, and fired four shots with the stolen gun.
Orly is Paris' second-biggest airport behind Charles de Gaulle, serving domestic and worldwide flights, notably to destinations in Europe and Africa.
A police source identified the man only as Zied B. and said he was known to the police for theft and drug offences.
"I said I didn't forgive him 'because you hurt a policeman'".
No-one else in the busy terminal was hurt, but thousands of travellers were evacuated and flights were diverted to the city's other airport. "Afterwards, we heard two shots", he told BFM TV.
A terrorist enquiry has been launched into the death of Ben Belgacem, who was last released from prison in November.
The French National Police tweeted that there was a "police operation" underway at the airport.
Once there, a prosecutor said, Belgacem threw a bag containing a gas can at soldiers before hurling himself at them.
Those soldiers were part of a military patrol stationed at the airport as part of the country's "Sentinelle" security program, which has deployed special forces troops at various sensitive locations throughout France following the series of deadly terrorist attacks that have struck the country since 2015.
The attack forced Orly to shut down, sent passengers fleeing in panic and trapped hundreds aboard flights that had just landed.
Traffic was jammed near the airport and people wheeled suitcases down the road. No explosives were found.
He was promptly gunned down in an airport terminal on Saturday morning, on the day that the Duke and Duchess of Cambridge were visiting the French capital. He was shot and wounded and taken into custody.
"Sentinelle" is the government's stepped-up security response to the January 2015 Islamist attack on the satirical weekly Charlie Hebdo which killed 12 people.