The softly-spoken 65-year-old confirmed that the defeat was his last match as a coach, and that he was going to work as a television pundit in his native Germany.
"I have a quiet life in front of me," a stoney-faced Hitzfeld told a news conference at which reporters were requested not to ask him about the passing of his brother.
"My job as a coach has finished here. I am proud of my career. I was very fortunate to coach some marvellous teams, and the Swiss national team.
"It has been a great honour ... and I am proud to say goodbye to the Swiss team with my heart full of emotions."
Hitzfeld joined the national team in 2008 and was seeking to lead Switzerland to their first appearance in the World Cup quarter-finals since 1954, when they reached the last eight on home soil.
The loss ended one of the most distinguished careers in European football.
The former player won two Swiss championships, seven German championships and three German Cups.
He also won the Champions League twice - with Borussia Dortmund in 1997 and Bayern Munich in 2001 - making him one of only a handful of coaches to lift the trophy with two different clubs.
Argentina were widely expected to progress without too much difficulty, but Switzerland's organisation in defence and intelligent play down the wings frustrated them for nearly two hours.
Lionel Messi, Argentina's talismanic forward, was kept at bay as Swiss defenders swarmed round him in numbers. But typically for the man able to change any game, he rode a tackle to set up Di Maria with a perfect pass to his right.
"We know that Messi, in one second, can decide a match. He has sufficient qualities to do that," Hitzfeld said.
"But then (there was) the pass to Di Maria and then the marvellous shot by Di Maria. It was certainly not easy to get past our goalkeeper. (Diego) Benaglio had a great match."
Hitzfeld was asked whether he regretted being too defensive against Argentina, but he said he had got the tactics right.
"You can, of course, surprise Argentina, but when Argentina have sufficient space, look at what happened in the last three minutes," he said.
"If we had attacked against Argentina we would have let in more goals than against France."
Hitzfeld said the game was emotionally super-charged.
"Today, this (emotion) was a huge dimension. It is a World Cup and you had the chance of getting to a penalty shootout and that of course creates enormous tension. I think we can walk tall and leave football with our heads held high."
- Sports & Recreation
- Angel Di Maria