Prior to the game, team spokesman Marco von Ah said Hitzfeld was "very sad" but was determined to stay with the team for its crucial game against the two-time world champions, which it lost after a goal in the 118th minute.
The team said 81-year-old Winfried Hitzfeld died in Switzerland after a long battle with blood cancer. It provided no other details.
"He wants to look forward. He wants everybody to do so and respect his private sphere," said von Ah.
A win over Argentina would have equaled Switzerland's greatest World Cup achievement, putting them in the quarterfinals for the first since it hosted the tournament in 1954.
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The 65-year-old Hitzfeld had said previously he would retire after the tournament. Prior to his postgame press conference, a Swiss official asked reporters not to ask the coach about his late brother saying he would like his privacy to be respected.
Several of his players, though, offered condolences after their emotional loss to Argentina.
Midfielder Gelson Fernandes said the coach did not mention his brother in the changing room but all the players were aware of the situation and fought extra hard to get Hitzfeld the win.
"He stayed strong. He gave us confidence and he was really, really strong," he said. "I am of course sad for his family and I wish him all the best in the future because he is not just a great manager, he is a great man.