The Americans fell behind to an early goal by Nani but second-half strikes from Jermaine Jones and Clint Dempsey put them ahead.
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"We have one less day to recover than Germany – they played yesterday and we played tonight," Klinsmann told BBC Sport on Monday night after the match.
"We have played a match in the Amazon while they have played in a better location where it is easier to travel around. Everything was done to help the big favourites so we have to do it the hard way now but we will."
The USA boss, who will lead his side into battle against the nation of his birth and the team whom he coached to third place at the 2006 finals, had nothing but praise for his players despite the heartbreak of conceding a late equaliser.
Klinsmann continued: "Obviously to lose a goal in the last second is unfortunate but it was an amazing game. We put in an amazing performance and there was nothing more we could have asked for from our guys.
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"Everything we have talked about and planned in recent weeks came into play during the game. We lost an early goal, which was unfortunate, but I felt we were in control of the game. We were setting the tempo and possession and could have had three or four goals.
"We have to build on those positives. We have another game that is like a final now but that is what the World Cup is all about.
"We scored two fantastic goals. The players absolutely played to their limits. In games like this you have to prove your quality and show the world that you are one of the best players in your country and we did that. It was an amazing job."
Germany and USA are in pole position at the top of Group G on four points, with Portugal and Ghana trailing with just one point apiece.
A draw in the last match in Recife on June 26 would see both teams progress, with Joachim Low's men as group winners due to their superior goal difference.