Angel di Maria scored in the 118th minute as Argentina claimed a 1-0 win over Switzerland in the round of 16 clash on Tuesday.
For Hitzfeld, it marked the end of a 31-year coaching career that included winning UEFA Champions Leagues with Bayern Munich and Borussia Dortmund.
"As for my future, I'm retiring from coaching so I can lead a quieter life," the 65-year-old German told reporters after his team's exit.
The Swiss coach was rocked by a personal loss a day out from the last-16 match, with the news his brother Winfried had died aged 82.
Hitzfeld was proud of his team, who were unlucky not to force Argentina to a penalty shootout in Salvador.
Lionel Messi set up Di Maria before Blerim Dzemaili almost equalised in the dying seconds, hitting the post with a header.
"We played the game we had to play, but you can see for yourself what Argentina can do when they have space," Hitzfeld said.
"If we'd gone out and attacked them, then maybe they would have scored more goals than France did (in Switzerland's 5-2 loss in the group stages).
"The idea was to stop Messi by putting three or four players on him, but when you're up against a team like that you can't just concentrate on one player, as Di Maria showed. It took a great finish to beat (goalkeeper Diego) Benaglio today.
"I'd like to congratulate my players for the way they reacted at the end. Switzerland earned the recognition of many people around the world and we can walk away with our heads held high."
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