Ziyed Ben Belgacem, 39, was killed after he put a gun to a soldier's head saying he wanted to "die for Allah".
Earlier on Saturday, he was involved in a shooting and then a carjacking.
His father told French radio Belgacem had phoned after the first attack to say he had "screwed up with a gendarme", before proceeding to Orly.
Belgacem is described as having been radicalised in prison, and was on a police watch-list.
His criminal record included convictions for armed robbery and drugs offences, Paris prosecutor Francois Molins told reporters late on Saturday.
Intelligence agents searched his home in Garges-les-Gonesse, north of Paris, for evidence of Islamist sympathies, but reportedly found nothing. However, traces of cocaine were discovered during the search.
The sequence of events
Early on Saturday morning, Belgacem was stopped at a checkpoint in Garges-les-Gonesse, north of Paris.
He fired at police with a pellet gun before escaping in a car that was later found abandoned.
Police say he then stole a car at gunpoint from a woman at Vitry, south of Paris. That car was later found at Orly airport.
He tried to seize a servicewoman's automatic weapon, put his gun to her head and said: "I'm here to die for Allah. In any case people are going to die."
He was then shot dead by two other soldiers.
A copy of the Koran was found on his body, Mr Molins added.
His father and brother were taken in for questioning in the aftermath of the attack. The brother remains in custody.
Speaking to Europe 1 radio on Sunday, the father said his son was not a terrorist, but someone who "never prayed and he drank".
"This is what happens under the influence of drink and cannabis," the father said.
He said he had received a phone call from Belgacem after his initial attack in which he said: "Dad, please forgive me. I've screwed up with a police officer."
Then he hung up saying he was on the motorway.
The BBC's Hugh Schofield in Paris says the picture is building up of a man on the criminal fringes of society, who also consorted with Islamist radicals.
Orly - located 13km (8 miles) south of Paris - is the capital's second-largest airport.
The attack came at a sensitive time. France has presidential elections starting next month and remains under a state of emergency.
The soldiers at Orly were part of Operation Sentinel - involving thousands of soldiers deployed to provide back-up to the police after the Charlie Hebdo attack in January 2015 and the Paris attacks of November 2015.