The 29-year-old shouted "Allahu Akbar" as he launched the gun attack, prosecutors claim.
It is believed the attacker - wielding a gun and a knife - was injured during a gun battle with security forces in central Strasbourg before he hijacked a taxi and escaped.
Chekatt was known to intelligence services before the shooting rampage - he was listed on a security watch list, had been in and out of jail in France and neighbouring Germany and Switzerland, and had been monitored for religious radicalisation.
"Terrorism has hit again," Paris public prosecutor Remy Heitz told reporters on Wednesday.
Chekatt had been imprisoned in Germany in 2016 and 2017 on theft charges, and was deported to France in 2017, a spokeswoman for Germany's BKA criminal police said.
The suspect is a 29-year-old French citizen who is known to French authorities as a radicalised Islamist, the spokeswoman said.
Germany's interior ministry said it has no information that Chekatt has an Islamist background.
The gunman fired his first shots just before 8pm local time just as the picturesque Christmas market in the historic city was winding down for the evening.
He engaged in two brief gunfights with security forces as he evaded a police dragnet.
Eyewitnesses told investigators that the gunman shouted "Allahu Akbar" as he carried out the attack.
France has said a "radicalisation" in Chekatt's "religious practices" was detected while he was in jail, but there were no signs that he was preparing an attack.
The suspect had evaded arrest just hours before the shooting when police raided his home on Tuesday morning in connection with a burglary and attempted homicide investigation.
Chekatt, who has 27 convictions, was not home at the time.
Mr Heitz said police found a grenade, a loaded gun, ammunition and knives.
Five people were detained and under interrogation as part of the investigation into an incident in Eckbolsheim, just west of Strasbourg, in August.
In that incident, a woman was stabbed when four men armed with guns tried to rob her at her home, it is claimed.
The three victims of the Strasbourg shooting include Anupong Suebsamarn, a 45-year-old holidaymaker from Thailand.
He is said to have arrived in France with his wife and another couple a day before he was killed.
Eyewitnesses said he was shot in the head and he was not responding to attempts to revive him in the immediate aftermath of the tragedy.
US President Donald Trump tweeted: "Another very bad terror attack in France. We are going to strengthen our borders even more. Chuck and Nancy must give us the votes to get additional Border Security!"
Strasbourg Mayor Roland Ries said there was no doubt that it was a terrorist attack.
France has deployed more than 600 security forces officers - including elite commandos and aerial units in helicopters - to search for Chekatt and guard sites amid fears of another attack.
It has raised its security threat to its highest level and strengthened its borders in case the suspect is still in the country.
However, officials have said it's possible that Chekatt, who was born in Strasbourg, has already fled across the border.
Four of his relatives were in custody for questioning, said Mr Heitz.
The anti-terror section of the Paris prosecutor's office, which is leading the investigation, has not yet revealed a possible motive for the attack.
The taxi driver whose car was hijacked told authorities that the suspect had suffered an arm injury during a gunfight.