Larossi Abballa was a 25-year-old Islamic extremist convicted earlier for jihadist activities and who was under investigation by French authorities.
An edited version of the Facebook video was disseminated on the ISIS-affiliated website A'maq in which the attacker threatens that Euro 2016 would "be like a cemetery."
The video showed Abballa smiling eerily after making the threat. He admitted to killing the police commander, Jean-Baptiste Salvaing, and Jessica Schneider, a civil servant who worked at a local police station.
The attacker pledged allegiance to the leader of ISIS and made a call for Muslims to carry out jihad on so-called infidels.
"Pledging allegiance to my emir, Abu Bakr al-Baghdadi, my first words are for the Muslim community: What happened to you? What kind of sentence did you receive? How did you get to this?" he asked.
The original Facebook video also showed the couple's 3-year-old son, a source close to the French investigation told CNN, although the attacker's Facebook page and original video have since been removed. A SWAT team was able to rescue the boy, the French Interior Ministry said.
The attack occurred shortly after 8 p.m. Monday. Abballa stabbed Salvaing first and then took Schneider and the couple's boy hostage in their home in Magnanville, Paris prosecutor Francois Molins said Tuesday.
The attacker targeted the couple because of their profession, Molins said.
Facebook video broadcast from scene
At 8.52 p.m., while in the couple's home, Abballa posted a 12-minute video to Facebook, Molins said.
During the Facebook broadcast, the child could be seen on a sofa behind Abballa, who told viewers he was not sure what to do with the boy, a source close to the investigation told CNN.
He also claimed in the broadcast that he was responding to a call from a senior ISIS leader for the terror group's followers in Europe and the United States to carry out attacks during Ramadan
Abballa told SWAT team negotiators that he was a devout Muslim and had pledged allegiance to ISIS weeks earlier, Molins said. After a standoff, the SWAT team raided the house and killed him, he said.
At the scene, police found knives and a hit list of targets, including celebrities, rappers, police officers, prison guards and journalists, Molins said. A Quran was recovered from the attacker's car.
Asked by CNN about the video, Facebook said it does not comment on any active investigations but said it works as quickly as possible to remove content that supports terrorism.
Conviction for jihadist activities
Abballa had been convicted and jailed in 2013 for being part of a recruiting network for radical Islamists to go to Pakistan and Afghanistan, Molins said.
Police had been investigating him since February on suspicion of links to a network recruiting jihadists to fight in Syria, Molins said. Although he was being monitored, no incidents had been flagged to authorities overseeing his case, he said.
He had also been known to proselytize for radical Islam.
Three people -- ages 27, 29 and 44 -- were taken into police custody for questioning in relation to Monday's attack, he said.
A'maq, the ISIS-linked agency, reported that an "Islamic State fighter" had carried out the killings in France.
'An abject terrorist act'
Speaking to reporters Tuesday, French President Francois Hollande said that the couple had been "assassinated by a terrorist."
"It was undeniably an act of terrorism, both because the perpetrator -- who was taken out at the scene, thanks to the quick reaction of the security forces -- wanted it to be recognized as an act of terrorism, and the organization he had pledged his allegiance to also claimed the attack."
French Interior Minister Bernard Cazeneuve described the killings as "an abject terrorist act." He said flags would be flown at half-staff for three days. Government workers observed a minute's silence at noon local time.
According to Magnanville Mayor Michel Lebouc, Abballa grew up in nearby Mantes-la-Jolie, which has a history of radicalization in its mosques. Magnanville is 45 kilometers (28 miles) northwest of Paris.
Monique Fohrer, a local resident, described the standoff Monday night as "a war scene."
"We didn't think we were in Magnanville. We thought we were in a film, a bad film," she told CNN.
Tournament held amid state of emergency
The killings come as France is on high alert for terrorist attacks
as it hosts the Euro 2016 soccer championship, which has been marred by outbreaks of violence between rival fans.
U.S. and British officials have both warned citizens about the potential terror risks of attending.
France has been under an official state of emergency since November's ISIS terror attacks in Paris
that killed 130 people. An ISIS cell connected to the Paris attacks then went on to strike in neighboring Belgium in March, killing 32 people
The same jihadist network responsible for those attacks sought to target the Euro 2016 tournament, ISIS terror suspect Mohamed Abrini told interrogators, according to a source close to the investigation.
The French government extended the state of emergency until the end of July to cover the Euro 2016 and Tour de France cycling race.
Law enforcement stretched
However, French authorities face a strain on resources in responding to the jihadist threat.
Following the Paris attacks, Hollande vowed to expand the government's counterterror resources in response to ISIS,
doubling the number of surveillance officers over five years to 10,000.
Intelligence officials have told CNN
that it takes 15 to 20 staff to monitor a suspect around the clock -- and France has about 11,000 people on its list used to flag radicalized individuals considered a threat to national security.
In comments Tuesday to CNN affiliate BFM-TV, national police spokesman Jerome Bonnet acknowledged the challenges police face with resources stretched by the terror threat, security for Euro 2016 and ongoing strikes and protests around the country.
"Police are fighting on several fronts," he said.
"They are tired, but they have fuel for their action. ... They love their job."
CORRECTION: Because of an editing error, an earlier version of this article erroneously described what a source said was shown on the video. The source said that the live video showed the man discussing the attack.