Mario Goetze put Germany in front in the 12th minute, with the hosts dominating much of the first half but wasting a raft of chances.
Mix Diskerud got the equaliser against the run of play in the 41st minute for the Americans, who then controlled much of the second half.
After Wood made it 2-1, U.S. goalkeeper Brad Guzan deflected a Sami Khedira header over in stoppage time to ensure victory for the visitors.
"It was a happy ending for us, to win against the world's number one team," said U.S. coach Juergen Klinsmann.
"It's something special and the players are really pleased. It was a balanced match with chances on both sides. But I think we deserved to win it, It was an important step for us."
Germany coach Joachim Loew said his side lost their way after the restart.
"We lost our strength in the second half and the Americans were more intensive," he said.
"You could see that we weren't able to respond to that."
It was the third match in three years between the United States, coached by former Germany boss Klinsmann, and Germany and their coach Loew, who was Klinsmann's assistant from 2004 to 2006.
The United States won a friendly 4-3 in 2013 before Germany won their World Cup group match 1-0 in 2014.
The U.S., who are tuning up for the Gold Cup they are hosting next month, came from behind to beat Netherlands 4-3 in a friendly in Amsterdam on Friday. Germany play Gibraltar on Saturday in a Euro 2016 qualifier.
"I don't know what happened to us in the second half," said Germany captain Bastian Schweinsteiger. "The Americans had one good chance in the first half and scored. Obviously we wanted to win this. Something like this shouldn't happen to us."
Schweinsteiger also blamed a lack of fitness for the disappointing second half after Germany started brightly.
"We really wanted to win this match even though it was a friendly and it was important for us," said American Alfredo Morales, one of several players who have played for clubs in Germany. "We beat the World Cup winners. That's quite a statement."
(Reporting by Erik Kirschbaum, editing by Pritha Sarkar and Toby Davis)
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