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Mix Diskerud then had a sight of goal for the visitors a few moments later, as he fired a powerful shot narrowly over the bar.
Joachim Low's side were in complete control and they had a chance to double their lead when Ozil threaded a pass through to Gotze, who then saw his low effort kept out by Guzan.
They came close again just minutes later as Gotze whipped in a cross from the left that was met by Andre Schurrle, but the Wolfsburg player could only flick his header straight into the gloves of the keeper.
Despite all of Germany's possession and chances, the visitors found themselves back on level terms just before the break, as Diskerud was picked out by Michael Bradley's pass, before slotting a composed finish past Ron-Robert Zieler.
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The goal provided a major boost to the USA and they were soon on the attack again, with Gyasi Zardes firing a powerful strike inches over the bar.
The teams headed into the break on level terms and when they re-emerged for the second half it was Jurgen Klinsmann's side who looked the more threatening.
They were pegging the Germans back for large spells, but they struggled to create any decent openings, until substitute Jordan Morris flashed a low effort inches wide of the far post.
The USA looked the more likely to snatch a winner and with eight minutes left to play Bradley almost scored a second, but after being found by DeAndre Yedlin inside the box, he saw his strike blocked by the feet of Zieler.
Germany failed to heed that warning, as four minutes later Wood collected a pass from Brad Evans just outside the box, before turning Shkodran Mustafi and firing a low strike into the far corner to give his side the lead.
The hosts almost scraped a draw in the final minute of stoppage time when Sami Khedira headed against the bar, but they could not find a way past Guzan, as the USA held on to record a famous victory.