Daniel Sturridge's wonderful goal had given Klopp's side control of the tie but Kévin Gameiro's equaliser just 17 seconds after the break changed the course of events.
From that moment Liverpool's confidence drained and Sevilla's domination became absolute.
Their fast, aggressive play was rewarded with victory thanks to further strikes from their captain, Coke.
It means that Liverpool will not feature in European competition next season and Sevilla progress to the Champions League instead.
Klopp was disappointed with the reaction to Gameiro's strike - a consequence of a succession of mistakes made by Alberto Moreno, the former Sevilla left back.
The first goal is not the problem,” he said. “What happened after that is the problem. We lost faith completely in our style of play, we lost shape and we were nervous in defence.
That shows our development is not as good as hoped. We have to carry on and use this experience, I am sure we will be strong. It is disappointment and frustration tonight.
“We are responsible for not being in the Champions League next season but now we have to use the time. We have to use it to train more and prepare more.
“It''s not about the size of the squad or how many players we have, it''s about using the time for training to get better, using this experience tonight and I promise everyone, we will use it. We will use it. We will come back stronger 100 per cent.
“The guys tried everything. We didn''t get it tonight, but we will come back.”
Klopp added: “I''m responsible for this performance too. No criticism allowed for my side. We couldn't reach our level, and it's my job to help the boys to see there's opportunity, to not feel the pressure too much. It's my job to help the team to react better. So, I can improve a lot.”
Unai Emery, the Sevilla manager, had instructed his players to “write history.” This was his third Europa League triumph in successive seasons.
“Maybe we are not the ones who should say, maybe others should analyse what we do so well,” Emery said. “I have my own theory, perhaps the level in Spanish football is very good, but competitive also. It has been proven Spanish teams compete passionately and in this circumstance they can win.
The result means, indeed, that Klopp''s own run extends. He has now lost five cup finals in a row.
“I know there are bigger problems in life,” he concluded. “But right now, it doesn't feel like it.” – The Independent
Original source: Reds will ‘come back stronger’ - Klopp