The Catalan giants hit the headlines at the start of the week when ex-president Josep Maria Bartomeu was among those arrested on Monday – he was then provisionally released under charges of unfair administration and corruption of business – as part of a police investigation into last year's 'Barcagate' scandal.
Lucas Ocampos had missed a chance to level on the night when his penalty was saved by Barca goalkeeper Marc-Andre ter Stegen with 17 minutes remaining and Sevilla had Fernando sent off before Pique struck.
Martin Braithwaite put Barca in front early in extra time and Luuk de Jong – who had been substituted – was shown a red card on the touchline for dissent after Sevilla were denied another penalty.
Barca head coach Koeman hailed the character shown by his side, who also beat Sevilla 2-0 in LaLiga on Saturday.
The former Netherlands boss said in quotes reported by Marca: "We always believed. We never gave up on the Copa. It's a question of mentality.
"I, as a coach, can't ask my team to do more than what I've seen tonight. We fought until extra time and it's incredible. I'm very happy with the team's work tonight.
"We deserved to go through. I think we were better over the two legs. We were the best team."
Koeman added in a week that will end with the Barca presidential election: "We managed to score three goals and could've scored more. You can't ask for more."