Clinton was sitting in the stands with dignitaries at Loftus Versfeld in Pretoria on Wednesday when forward Landon Donovan got a winner in added time to send the U.S. into the last 16.
"I lost my voice yesterday. I was very diplomatic until we scored," Clinton said in an interview with a roundtable of journalists on Thursday.
"When that sucker went in there, I said, 'Thank God for overtime'," he added, having joined in the celebrations after a pulsating 1-0 win over Algeria on Wednesday that left the U.S. top of Group C above favourites England.
"Both the United States and Algeria played that game with both their minds and their hearts in the right place."
"In the locker room, all they talked about was how they played as a team. They were a happy crowd yesterday."
The former U.S. leader said while the world's football powers do not see the United States as reaching their level, there is a grudging respect for the battling Americans.
"Argentina and Brazil think that we are better at this then we are," he said. "Others have been doing it longer. That is OK. I think they believe that we are serious about it now."
"It would be really good for America if we could do it. I think it would be good for soccer worldwide if we could do it."
"The real trick is not to ask people to give up the particularities of their identities but to give them a place to put it that is not destructive," he said.
