Daniel Sturridge's exquisite first half goal would have been worthy of winning any UEFA showpiece but Jurgen Klopp's side failed to capitalise and went to pieces in the second half.
Kevin Gameiro scored just seconds after the interval as the Reds were exposed defensively before Coke grabbed two in the space of six minutes to end Liverpool's dream of a return to the gloried European nights of yesteryear and also a place in next season's Champions League.
In the build-up Liverpool's players were at pains to stress how much belief Klopp has given them during his seven months in charge and how they had learned from the experience of losing February's Capital One Cup to Manchester City.
iverpool had three penalty shouts in the first half which began withstanding some early Sevilla pressure before settling their nerves to control the last 20 minutes.
Roberto Firmino's flick past Daniel Carrico hit the arm of the centre-back but Swedish referee Jonas Eriksson ignored appeals.
The two claims against Adil Rami and Grzegorz Krychowiak were not as strong but highlighted Liverpool's increased dominance in the final third.
Emre Can had a shot palmed away by David Soria but it was Sturridge who was involved in their most threatening moments, directing a far-post header back across goal and then having a shot blocked by the goalkeeper from Adam Lallana's reverse pass.
Firmino and Philippe Coutinho combined to tee up Sturridge on the left of the penalty area.
The angles looked wrong for the predominantly left-footed England striker but he caressed a sumptuous shot with the outside of his stronger foot which arced around Soria and inside the far post.
Sturridge nearly turned in another cross from the tireless right-back Nathaniel Clyne before the break but the sight of Klopp sprinting to the dressing room suggested he knew there was still plenty of work to do - especially if Gameiro's overhead kick just wide of the post from a corner was any indicator of the level of threat from the Sevilla striker.
The Frenchman needed just 18 seconds from the second half kick-off to prove that point as Mariano Ferreira exposed Liverpool's left side and Gameiro diverted home from close range and only Kolo Toure's perfectly-timed tackle prevented him making it 2-1 seconds later.
Only Gameiro's mis-hit shot gave Simon Mignolet a chance to save after Alberto Moreno switched off at a throw-in as Liverpool found themselves under the cosh and their resistance was broken by a rapid interchange of passes which saw Coke fire home in the 64th minute.