Juventus held onto their 3-0 first leg lead to progress to the last four by blunting Barca's star-studded attack in a 0-0 draw yesterday.
Neymar left the field in tears, but around the Camp Nou there was appreciation for Barca's valiant effort by the near 100,000 crowd.
"It's a sad night, but we tried until the last minute and that was recognised by the fans at the end," said Enrique, who will step down at the end of the campaign.
"It is a joy to belong to a club like this."
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Barca travel to face eternal rivals Real Madrid in El Clasico on Sunday realistically needing to win to maintain their hopes of a third straight La Liga title.
Madrid lead by three points at the top of the table, but also have a game in hand.
"Barca have a team built to compete, to win trophies and to face every challenge.
"It is clear we are very hurt to be eliminated from a competition that excites us so much, but now comes a key game against the leaders and a direct rival.
Barca will be without Neymar at the Bernabeu through suspension and the Catalans will need Lionel Messi back in top form as the Argentine was off target with the only two clear chances the hosts created yesterday.
"We had 13 or 14 shots, but I think only one was on target, which are the ones that tend to do damage," added Enrique.
- Juve believe -
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Juventus are now in the last four for the second time in three years under coach Massimiliano Allegri.
"The self-belief of the team has grown a lot," said Allegri.
"There are only two players who started tonight who started the final in Berlin.
"That says a lot of how well the club has recruited.
"We are desperate to go as far as we can and achieve something extraordinary."
Barca defender Gerard Pique recognised that Juventus had deserved to go through over the two legs.
"Overall they were the better team and deserved to go through," he told BeIN Sports Spain.
And Pique also called on the Barca players to reward the fans' support at full-time with victory in the Clasico.
"That the fans were still singing needs to be remembered. I think it is one of the few times I have seen the Camp Nou like that in a defeat. It moved me and makes me proud," he added.
"We will go to the Bernabeu to compete and try to win.
"In the last few years it has been good to us. We are not in our best moment, but we can win.