Graham Arnold's side claimed the three points through a 76th-minute goal by Zidane Iqbal and the loss, Indonesia's second in a row after a 3-2 defeat by Saudi Arabia on Wednesday, means the country's hopes have been extinguished.
"I think if you look at the game, we were by far the better team," said Kluivert. "I'm very disappointed about the result and not only me but the whole country, the players, the staff.
"We worked hard, playing against Saudi and then three days later, it was a great recovery to play another tough game. We performed fantastic.
"I'm very proud of the players, who showed their heart, their courage but unfortunately the result again was not on our side while creating chances and playing very good football. With one action you lose the game. We are devastated."
Indonesia were looking to qualify for the World Cup for the first time since the country's independence from the Netherlands in 1949, and their progress to the fourth round of Asia's preliminaries was the country's best performance to date.
However, an inability to score in open play ultimately hurt a team built largely on players from the country's Dutch diaspora, with both of Kevin Diks' goals in the Saudi Arabia loss coming from the penalty spot.
"We worked hard as a team, medical staff, local staff, players, everyone really worked hard for this," said Kluivert. "All the results that we had, good and bad results, I think we grew as a team, individually and collectively.
"The World Cup dream slips away through your hands. The disappointment, of not only me but the whole country of Indonesia, is really tough. It's really tough."
Kluivert dismissed suggestions the team's lack of experience had cost them against more wily opponents.
"They're not too young," he said. "The way they played was fantastic, that they can play like this against Saudi, against Iraq, who are 58th and 59th in the world ranking and we are 118.
"As a head coach, I'm very proud of them, if you can show your heart like this. Unfortunately the goals are not coming when you create the chances and you play good football."