Ousmane Dembele, Gerard Pique and Martin Braithwaite were all on target for the Catalans as their 2-0 deficit from the first leg was wiped out.
Dembele broke the deadlock in just 12 minutes, giving the hosts a headstart, but Julen Lopetegui's side held firm in the match.
Lucas Ocampos then had the chance to pull one back for the visitors from the spot in the 74th minute and put the contest to bed, but saw his penalty attempt saved by Marc-Andre ter Stegan.
That gave Barcelona some much-needed impetus and, following some lengthy spells of sustained pressure, Gerard Pique headed in their second goal in the fourth-minute of stoppage time just after Fernando's sending off.
The game headed into extra-time and substitute Martin Braithwaite completed the turnaround in the 95th-minute, with Luuk de Jong also seeing red for the visitors later.
Barcelona are now in their seventh League Cup final of the last eight years and will face either Levante or Athletic Bilbao in April's showpiece clash. Here are the hits and flops from the Camp Nou thriller:
Hit: Gerard Pique (Barcelona)
Too often the scapegoat during last year's disastrous season for Barcelona, Gerard Pique kept fighting to prove himself and retain his position in the side.
Even though the veteran Spaniard had offered no great shakes so far this season, he continued to show his worth to Ronald Koeman, and tonight, his persistence paid off.
Like he's so often done in his prime, Pique popped up with a late header which kept the Catalans in the game, thumping the ball home from close range in the fourth minute of stoppage time to bring his side level on aggregate.
That was Pique's first goal of the season and he couldn't have timed it any better as it set Barcelona on their way to a historic victory.
Flop: Lucas Ocampos (Sevilla)
He had one job to do and Ocampos managed to screw that up too.
Silent for most of the match, the Argentinian targetman had the opportunity to put the contest to bed from a penalty that he won himself.
At the time, the game was still only 1-0 in favor of the hosts and Ocampos dispatching the penalty would've all but ended Barca's night.
However, the 26-year old stood up and smashed a limp effort which was easily saved by ter Stegan, keeping the tie in the balance and letting Barca heave a huge sigh of relief.
Barcelona eventually managed to turn the tie around, something which wouldn't have been possible had Ocampos just taken his chance. This will surely haunt the player for the next few days.