In one of the most entertaining games of the tournament so far, it was all square at half time
after Angel Di Maria cancelled out Antoine Griezmann's penalty, before Gabriel Mercado's goal shortly after the restart gave Argentina the lead.
That was until goals from Benjamin Pavard and a brace from Kylian Mbappe in the space of 11 minutes put Les Blues back in the ascendancy, with Sergio Aguero's late header little more than consolation for La Albicseleste.
Now reaching the last eight for the second successive World Cup, Deschamps admitted post-match his side's mentality was the most impressive aspect of their performance, especially after going behind so early into the second half.
As quoted by ESPN, he said: "We were up against a highly experienced Argentina team but we showed we were there and the team showed they had character.
"When we were 2-1 down it was not easy, but we kept fighting. The mentality in this group is excellent. As long as you have time, even if you have a highly experienced team in front of you, I know that the French team are ready to face it."
Considered by many as one of the favourites to win in Russia, especially after their attacking display in their last 16 win, Deschamps acknowledged the emotions behind France's win, as they now prepare for only their third World Cup quarter final since 1990.
Speaking to TF1 (as quoted by ESPN), he stated: "It was emotional. I am very happy for the players and to have qualified. I feel happy.
"As I told the players before the match, they have been working towards this for months, they were ready for a game like that. They needed to answer, and the response was a strong one."
The prize for beating 2014's runners up in Kazan is a clash against Portugal's conquerors Uruguay in Nizhny Novgorod on Friday, with a semi final place in the World Cup on offer for the winner.