Acting on a tip-off from a local woman who had seen a man answering to Cheat's description in Neudorf, the district where he grew up, police sealed off the area while a helicopter equipped with thermal cameras flew overhead.
At about 8.00pm three police officers spotted him and tried to arrest him but he fired at them, according to Interior Minister Christophe Castaner, who arrived in the city during the evening.
"They immediately responded and neutralised the assailant," he said.
Subsequent tests have established that the man was Chekatt, the Paris anti-terror prosecutors' office announced later.
Some 700 members of the security forces had been mobilised to track him down after Tuesday's attack, during which he was wounded in the arm in a first shootout with police.
Germany also issued a warrant for him, in case he had crossed the border into his territory.
French president thanks security forces "Our commitment to fight terrorism is total"
Merci à l’ensemble des services mobilisés, policiers, gendarmes et militaires. Notre engagement contre le terrorisme est total.Emmanuel Macron (@EmmanuelMacron) 13 décembre 2018
IS claims killer as soldier
The Islamic State armed group issued a statement claiming Chekatt as "one of its soldiers".
Chekatt, 29, was born in Strasbourg and had been found guilty of various crimes 27 times in France, Germany and Switzerland.
He is reported to have become radicalised while in jail and to have stuck up a poster of the late Al-Qaida chief Osama bin Laden in his cell during one of his prison terms.
A plot to attack the Strasbourg Christmas market, which is visited by some two million people a year, was foiled in 2000.
It has been closed since Tuesday's attack but Castaner announced that it would reopen on Tuesday.
France remains on its highest level of terror alert.