The German side won 4-3 in Cincinnati at the TQL Stadium under a blazing sun, with the team's reserves watching the first half from the dressing room and the second under umbrellas on the bench.
"I'm sweating like I've just come out of a sauna," said Kovac, who had already complained about the expected conditions on Friday before the Group F clash with the South Africans.
"It was tough going for both teams, but the opposition are used to it. We certainly didn't give our best performance, but that wasn't possible today."
Dortmund posted a photo on social media of their substitutes watching the first half on television from within the dressing room.
"Our subs watched the first half from inside the locker room to avoid the blazing sun at TQL Stadium — never seen that before, but in this heat, it absolutely makes sense," wrote the German club on social media platform X.
Temperatures were at a high of 32 degrees Celsius in Cincinnati on Saturday, for the game which began at 12:00 pm local time.
Kovac said on the eve of the game there was no shelter on the bench from the sun, and warned media to bring sunscreen, hats and water.
Other teams have also complained about the summer heat in the United States, especially for matches kicking off in the early afternoon.
Atletico Madrid's Marcos Llorente said his "toenails" hurt because of the conditions at the Rose Bowl in Pasadena for his team's opening clash against Paris Saint-Germain on June 15.
Find more details on group standings, match schedules, and competition statistics.
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