The U.S. held firm for 90 minutes but fell down 2-0 in extra time. Despite fighting back to bring the game within one goal, the equalizer failed to come from more than one decent opportunity.
"At the end, you need a little bit of luck that is going your way, and today we didn't have that," Klinsmann said. "But when you look at the four games we played, we did a very very good job."
Klinsmann, who is under contract until the conclusion of the next World Cup in 2018, said the Americans' performance on Tuesday was something the country can stand behind.
"Obviously it's a bummer; we were so close," Klinsmann said. "I think we can all be very, very proud of this team. When you push Belgium to the limit or even beyond, that is something to be proud of.
"It could have gone either way. We had our chances as well. Things open up [in extra time] because the legs get heavy, and they push us to the limit. We had our chances to win it, but we didn't put it in, they put it in."
Goalkeeper Tim Howard kept the U.S. in the game, making a record-setting 16 saves.
"What Tim played tonight was phenomenal," Klinsmann said. "He had an absolutely amazing match and you have to give him the biggest compliments in the world.
"This was definitely an amazing goalkeeping performance. He should be very proud of himself and we are proud of him."
Klinsmann made a surprise move before the match by deciding to sit Kyle Beckerman, who had started all of the group-stage matches, and instituting a fearless 4-3-3 formation.
"We knew that one of the keys in this game would be the midfield battle," Klinsmann said. "Their midfielders were running the show. We weren't afraid of them, but we thought we could control them. The longer the game went on, the better we came into the game.
"They were all at their limits. We knew sooner or later that the team would hit a wall. We needed to wait with the third sub, but all of the players went beyond their capabilities. They should take a lot of positive stuff back home after this World Cup."
That third sub was Julian Green, the youngest player on the U.S. squad at 19, who made his World Cup debut as a substitute in extra time and made an immediate impact by scoring the U.S.' lone goal.
"Julian was growing in a fast speed the last seven weeks. We guided him through that process," Klinsmann said. "I knew that the moment would come today, but the injury of Fabian [Johnson] and issues in other positions delayed it. We knew that he was ready. I told him before the game to watch the No. 2 and to keep an eye on him. It was phenomenal how he came in and scored to get us back in the game. But it's fun to watch that kid grow."
The Americans advanced from a difficult group that included Germany, Portugal and Ghana to reach the knockout rounds of consecutive World Cups for the first time. Four years ago, they were also eliminated 2-1 in the round of 16, with the winning goal coming in the third minute of extra time against Ghana.
Although the team has been eliminated in Brazil, Klinsmann said he believes the future is bright for U.S. Soccer.
"We found ways to introduce new, young players and we've done a lot of work," he said. "We are excited about some players coming through the ranks. We are excited about building the next Olympic cycle. The way the fans embraced the team and the sport will only continue to grow; especially with the competitions we have ahead of ourselves. The Gold Cup, Olympics and Copa America in the United States. There's a lot to build on going forward.
"You hope that this team takes this experience and understands what this intensity and the level means. The demands, and it's not only training or playing a game on the weekend, it's about the lifestyle. They learned a tremendous amount and gave everything they had and you meet teams like Belgium and Germany and maybe have a little bit more quality.
"The talent gap is difficult to discuss. We've done well to recruit a lot of good talents and we need to have those competitions to make them grow. They fly home in a couple of days and will realize this is beyond their limits. This is not the Champions League. That experience will help them and we will simply keep building."
Fans who had made the trek south of the equator chanting, "I believe that we will win!" could hardly believe the Americans lost, extending a World Cup winless streak against European nations to nine games over 12 years.
The crowd of 51,227 at Arena Fonte Nova appeared to be about one third pro-U.S., with 10 percent backing the Belgians and the rest neutral.
Back home, millions watched across the United States in offices, homes and public gatherings that included a huge crowd at Chicago's Soldier Field.
"It's a bummer for us, ending on the losing side after a game of 120 minutes," Klinsmann said. "It was a real thriller. We had enough chances to equalize the game or put it away even earlier. It was a game that just went to the extreme.
"We are very proud of our team and they made their country proud. You have to swallow it for a second, but after a little break, you have to move on."
Klinsmann: It could have gone either way
United States head coach Jurgen Klinsmann praises his team's performance against Belgium and discusses the progress they have made.
Play of the day: Julian Green
United States forward Julian Green struck gold with his first ever touch in a World Cup game, a stunning volley to get the U.S. back in the game against Belgium.
Klinsmann Cam: Belgium vs U.S.
Check out the ups and down of Jurgen Klinsmann during the heart-breaking defeat to Belgium in the round of 16.
U.S. bow out in valiant fight against Belgium
Late drama in extra time saw the United States' Julian Green pull a goal back, but strikes from Belgium's Kevin De Bruyne and Romelu Lukaku were enough for a 2-1 win.
Bradley: Unlucky not to get the equaliser
United States midfielder Michael Bradley discusses how close his team came to levelling the match against Belgium.
Howard: It's heartache, it hurts
United States goalkeeper Tim Howard reacts to his team's heartbreaking loss to Belgium following a record 16-save performance.