Goals from the much-maligned Alvaro Morata and Mikel Oyarzabal finally settled an eventful contest in Copenhagen that had ended 3-3 after 90 minutes.
Croatia had taken an unexpected early lead when Spain goalkeeper Unai Simon allowed a backpass from Pedri to squirm into the net.
Spain responded superbly through Pablo Sarabia, Cesar Azpilicueta and Ferran Torres but Croatia hit back with two goals in the last five minutes of normal time from substitutes Mislav Orsic and Mario Pasalic.
Croatia carried their momentum into extra time but Simon made a fine save and Spain ultimately went on to secure a place in the last eight.
"It was a really intense game and I have experienced some similar ones, but it really had everything today. I think the strange thing is that we got a second chance and that doesn't usually happen," Enrique said.
"What really struck me was the final five minutes of extra time, the fans stood up and started applauding and they were paying tribute to the great effort and commitment of all of the players who had competed and I think it was a lovely gesture from the fans.
"It's about recognising the spectacle that both teams put on today."
He added: "I think the only mistakes we committed in the game were in the final 10 minutes when we were leading 3-1.
"I think we should have continued playing the way we had been playing beforehand and not deciding to go to long balls and play a defensive game because that's not the game I want to play, even if sometimes it is appropriate.
"I'm really pleased that we got a second chance and were able to win 5-3 and I think we did play out the end of that game the way we need to, which is playing and defending with the ball."
Croatia manager Zlatko Dalic felt his side tired after an energetic first half and was left to lament crucial mistakes as the World Cup finalists exited the tournament.
"I think we spent a lot of energy in the first half and that did not give us a great result, so I believe that we spent lots of energy at this point and then Spain did also have their own chances which they did not use," Dalic said.
"Then when it comes to the goals, we made three major mistakes. Spain is a great team that definitely will use any chance and will definitely utilise any fail in our system, but I think when we missed the chance at 3-3 in extra time, that was the determining one.
"If we would have scored a goal it would have been completely different, but congratulations to Spain and I also would like to congratulate my boys for what they have managed to achieve today and what they gave in this match."