Swept aside in the first half yet trailing only 1-0, the Spanish club equalized 18 seconds after the restart with Kevin Gameiro's goal from close range and Coke striking twice in the 64th and 70th minutes.
The third goal was furiously disputed by Liverpool after an assistant signaled an apparent offside against Coke, then lowered his flag.
Liverpool had led on Daniel Sturridge's beautifully curled shot with the outside of his left foot in the 35th. Defeat leaves Liverpool with no European football next season and dealt coach Juergen Klopp a fifth straight loss in a cup final.
Victory in the second-tier competition, for the fifth time in 11 seasons, lifts Sevilla into the Champions League group stage next season. It also ensures Spain will sweep Europe's club competitions for a third straight year, ahead of the Atletico Madrid-Real Madrid final of the Champions League on May 28.
Sevilla turned around the match magnificently to rock a Liverpool team that was rampant as the first half progressed. Still, two Sevilla goals exposed tackling and positional errors by its former player Alberto Moreno on the left side of Liverpool's defense.
The game changed on Sevilla catching Liverpool cold in the second half after Moreno's poor header gifted possession to Mariano Ferreira.
The right-back then broke too easily through a Moreno tackle to pass the ball across the goalmouth for the unmarked Gameiro to score. Gameiro should have scored in the 60th, when left alone by Moreno to hit a bouncing shot that goalkeeper Simon Mignolet saved, but Liverpool's reprieve was short.
Coke soon struck a sweeping shot from the edge of the penalty area, after Vitolo played two return passes and broke through a tackle. Coke then exposed Moreno's defensive space to fire low into Mignolet's goal.
Liverpool had led when its three forwards linked together for the first time. Brazilians Roberto Firmino and Philippe Coutinho passed across the Sevilla defense to find Sturridge to step forward and strike a sweet shot.
For once, the danger was not from Liverpool's right flank where defender Nathaniel Clyne had tormented left-back Sergio Escudero.
A deep chip in the 11th picked out Sturridge for a header that Daniel Carrico cleared from the goalmouth, and in the 45th the fit-again forward could not connect in front of goal on a slick cross.
Liverpool thought it had a two-goal lead in the 39th from a corner won by Clyne exploiting huge space. Dejan Lovren's strong header found the bottom corner of the net but goalkeeper Soria was distracted by Sturridge in an offside position.
After a tentative start, the high-intensity pressing game Klopp demands seemed to unsettle the two-time defending champion. Compared to Klopp's calm, hands clasped behind his back, his players tore into Sevilla seeking a deserved second goal.
Liverpool could have been awarded a penalty when the ball struck a Sevilla player's hand or arm for the third time â midfielder Grzegorz Krychowiak's left hand brushing the turf to divert a pass from Sturridge.
Sevilla's danger man Gameiro barely showed until winning a corner in the 30th that eventually found him for a scissor kick with his back to goal that flashed just wide.
Liverpool fans far outnumbered Sevilla supporters in the 35,000 crowd and fights broke out 30 minutes before kickoff in a poorly divided area behind one goal. Only then did a line of police and stewards stand between the two groups.