Germany won 1-0 in Ulsan thanks to a first-half goal from Michael Ballack.
Arena's side had taken several famous scalps on their way to the last eight, beating Portugal and Mexico.
Arena said: "I was really proud of our team, we were a little unlucky.
"Oliver Kahn came up with some great saves and kept them in the match. He was probably the difference.
"They are a big team, a strong team and they are very dangerous from free kicks.
We never once got played off the park out here and can go home with our heads held high
"In the match we had a lot of chances, the Germans had two and they converted one - that was the difference.
"We showed the world we can play. We haven't arrived but there's a bright future for the game in the United States."
USA keeper Brad Friedel, who enjoyed an outstanding tournament, agreed with his manager.
"We never once got played off the park out here," he said. "We can go home with our heads held high.
"They can play games like this, when maybe they are not the best team, and still win. That is why they are so good.
"Their man of the match was their goalkeeper, Oliver Kahn, who was fantastic.
"We had a hard time in the first-half but in the second we dealt with them."