La Roja dominated the first 45 minutes of the Group E contest, boasting more than 80% of the possession, and they took the lead in the 11th minute of the match through Alvaro Morata.
Japan turned the game around in the early stages of the second half, though, levelling through Ritsu Doan before Ao Tanaka scored a second for Hajime Moriyasu's team.
Spain still managed to advance to the round of 16 as Group E runners-up, with the European nation set to take on Morocco on Tuesday.
Enrique said that his side "have nothing to celebrate" despite securing their position in the last-16 stage of the competition.
"In football you deserve it or not and we didn't deserve it," Enrique told reporters. "I'm not happy at all. I would have liked to be on top and win but in five minutes Japan scored two. We were dismantled.
"We didn't have any danger in the first half and then at half-time I told them to be cautious because they had nothing to lose. We collapsed and they could have scored two more goals. I'm not happy at all.
"I never celebrate defeats so we have nothing to celebrate. We have qualified but I have nothing to celebrate."
Enrique also revealed that he was not aware that his side were down in third spot in the group for a brief period in the second half when Costa Rica took a 2-1 lead against Germany.
"We were knocked out for three minutes?" said Enrique. "I was not paying attention to the other match, when did that happen? I didn't know this. I'm not happy because we were defeated by Japan. If I had found out [Spain were going out] I would have had a heart attack."
Meanwhile, the head coach also commented on Japan's winning goal, which was controversial, as the ball appeared to be out before it was delivered into the penalty box.
"From my angle the ball was clearly half out," he said. "More than that I could not say because of the speed. I was concerned about scoring but there was always a possibility that it was going to be out.
"If it was not out and it was not a goal I would not have been disappointed but in the end it was a goal so that was great."
Spain finished on just four points in Group E, having beaten Costa Rica 7-0 and drawn 1-1 with Germany ahead of their contest with Japan on Thursday night.