Barcelona won a back to back double of La Liga and Copa del Rey
Luis Enrique's side claimed the trophy for the 28th time despite having Javier Mascherano sent off in the first half.
Jordi Alba and Neymar scored for Barcelona, who can boast back-to-back doubles after winning La Liga.
Sevilla had Ever Banega dismissed in stoppage time at the end of 90 minutes and Daniel Carrico at the end of extra time as they finished with nine men.
Jordi Alba celebrates his first goal of the season, which won the Copa del Rey
Mascherano was given a straight red card in the 36th minute for dragging down Kevin Gameiro by his shirt as the Frenchman raced towards goal.
Five minutes into the second half, Barcelona defender Gerard Pique produced a vital deflection to divert a long-range effort from Banega onto the post and, after an end-to-end second half, Banega was sent off in stoppage time for chopping down Neymar on the edge of the area.
Barcelona broke the deadlock seven minutes into extra time when Alba sprinted to meet Lionel Messi's fine crossfield pass and finished first time across goalkeeper Sergio Rico and into the bottom corner.
Carrico was dismissed in stoppage time at the end of extra time when he received a yellow card for a foul on Messi and another for his angry protest.
Messi then fed Neymar for the Brazilian to add a second with a confident finish.
No backward step for Barcelona
Barcelona refused to take a negative approach and, even when Mascherano was dismissed, coach Luis Enrique kept his side on the front foot by resisting the temptation to replace one of his forwards with a natural centre-back.
Instead, he deployed Sergio Busquets as a makeshift centre-back until half-time, then sacrificed midfielder Ivan Rakitic to allow defender Jeremy Mathieu to partner Pique at centre-back, allowing the attacking trio of Messi, Suarez and Neymar to stay on the field.
That was until 12 minutes into the second half, when Suarez pulled a hamstring and was forced off, but Barcelona still held their own despite Sevilla taking the initiative with their extra man.
Despite his reputation for being combative and short-tempered, it is the first time in his career Mascherano has been sent off twice in a single season.
It was also Barcelona's first sending off since October, when Mascherano was dismissed against Eibar, since when 13 opposition players have received red cards against Enrique's side.
Referee Carlos del Cerro Grande shows Javier Mascherano his second red card of the season
Barcelona's triumph ended a remarkable run in cup finals for their opponents.
Sevilla have earned a reputation as Spain's cup final experts, having not lost a final since 1962 apart from in one-off games in the Uefa Super Cup and the Spanish Super Cup.
Since losing to Real Madrid 54 years ago in the Copa del Rey final, Sevilla have won on their only other two appearances in the final, beating Getafe in 2007 and Atletico Madrid in 2010.
And in Europe they have triumphed in all five finals they have contested in the Uefa Cup and Europa League.
They have won the Europa League for the last three seasons - including last week's victory over Liverpool in Basel - to establish coach Unai Emery as one of the most successful in Europe on big occasions.
That success has brought him to the attention of other clubs, with Everton the latest to be linked with the 44-year-old after sacking his fellow Spaniard Roberto Martinez.