Sevilla was a bridge too far for Liverpool on their memorable Europa League journey.
Daniel Sturridge's superb swerving effort on 35 minutes saw the Premier League outfit clasp one hand on the trophy, but Kevin Gameiro, within a minute of the restart, levelled up for the Primera Division side.
Coke fired the Rojiblancos ahead not long after the hour mark, before giving his side a two-goal cushion on 70 minutes as Unai Emery's boys held on to maintain their impeccable recent record in Europe's secondary competition.
Understandably, a number of Liverpool players were downbeat at full time, but midfielder Emre Can looked particularly inconsolable.
Meanwhile, Klopp and chief executive officer Ian Ayre - who will leave the club next summer - shared a warm embrace when the runners-up went to collect their medals.
The defeat represents the second time that Klopp's boys have lost a Cup final this past season, having gone down to Manchester City on penalties in the Capital One Cup back in February.
But it's still been a hugely encouraging first season for the Reds under the German and few would back against them winning a trophy next term.