(Reuters) - Beaten but unbowed, the United States exited the World Cup after losing to Belgium on Tuesday, their disappointment to be going home offset by optimism for the future.

The Americans went down 2-1 in an extra-time thriller that ended with both teams' players running themselves to the point of exhaustion after one of the best matches of the tournament.

"Obviously it's a bummer for us to end on the losing side (but) it was a game which gave everything to the fans, to the crowd," U.S. coach Juergen Klinsmann told reporters.

"It was real drama, a thriller, we had enough possibilities to equalize the game or even put it away earlier. It was a game that just went to the extreme."

Klinsmann said he could not have asked for any more from his unheralded players, who defied the odds just to make the round of 16 then held on bravely against a skilful, well-organized Belgian team.

The Europeans attacked the United States from the outset, firing a barrage of shots, but were unable to find the net until extra time when Kevin De Bruyne and Romelu Lukaku scored.

"Every player was at his limits on the field. We knew sooner or later they would hit the wall," Klinsmann said.

"All the players today went beyond their capabilities. I told them in locker room they can take a lot of positives back home after this World Cup.

"We worked tremendously over the last few years and found ways to introduce new young players into our program and develop our game."

Klinsmann, who also serves as technical director of U.S. Soccer overseeing the development of the next wave of young players, said the team's performances in Brazil would provide another boost to the sport in North America.

"We must now swallow it and develop very quickly a young group. We have (2016 Rio) Olympics next and the special Copa America in the U.S. (in 2016)," he said, referring to an extra regional tournament to celebrate the centenary of South American football confederation CONMEBOL.

"We take a tremendous lot away from this experience. We grew a lot. We know we can play eye-to-eye with other nations. We know how important our young players are to us and now we can build to the next cycle."

BRILLIANT SAVES

U.S. goalkeeper Tim Howard, who made a series of brilliant saves to keep the Americans in the match, was heart-broken by the result.

"We played so well, it stings that much more. It really hurts to lose although Belgium played well.

"My own performance doesn't matter. I signed up to get my face in front of the ball. If I had no saves or 20 saves, it doesn't meant anything less.

"We dreamed but fell short of our dream. But we got out of the 'Group of Death'. And then we played a top team like Belgium. Gosh we were nearly there."

Midfielder Michael Bradley, who floated the ball over the Belgian defense for substitute Julian Green to score in the second half of extra-time and give the U.S. team hope of forcing a penalty shootout, said the match could have gone either way.

"When you get to this point, these games are always about a play here, a play there, chances going both ways," he said.

"We kept playing, kept fighting. probably unlucky not to get the equalizer at the end."