The incident comes as France remains on high alert following a series of jihadist attacks that have claimed over 230 lives since January 2015.
Prosecutors said they had opened an anti-terror investigation.
Flights were halted at Orly for several hours, with incoming flights diverted and thousands of passengers stranded at Paris' second-largest airport, but normal service resumed mid-afternoon.
France goes to the polls on April 23 in the first round of a two-stage presidential election in which security is one of the main issues on voters' minds.
He said the suspect was also linked to an attack on police just north of Paris today in which an officer was slightly wounded.
At the airport, the 39-year-old man knocked the soldier to the ground and grabbed at her rifle, a senior military source said.
The two other soldiers with her then opened fire, killing him, the source said.
No-one else was injured in the incident.
According to an eyewitness who spoke to AFP, the man had at one point held the female soldier hostage.
"I heard cries and I saw the soldiers who were holding a person at gunpoint."
The incident took place at around 8.30 AM in the Orly-Sud terminal. Around 3,000 people were evacuated from the building.
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.
He was thought to have stolen a car and driven to Orly, Le Roux said.
Sources close to the investigation said the man had committed several armed robberies and was investigated for radicalisation in 2015, although he did not feature on the list of those thought to pose a high risk.
Franck Lecam, a traveller bound for Tel Aviv, told AFP he heard "three or four shots" as he was queueing to check in.
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