Having lost the first leg at Sevilla, 2-0, Ronald Koeman's men knew they needed to score two goals without reply to force extra time at the Camp Nou, and instead did one better to win the tie on goals from Ousmane Dembele, Gerard Pique and Martin Braithwaite.
Barcelona stated their intent after just 12 minutes through a spectacular Dembele strike from outside the box. The France international took advantage of Sevilla defenders standing off him and curled and unstoppable strike high and inside the near post.
The Blaugrana took the lead into half-time and nearly pulled even in the tie in the 67th minute when an outrageous jumping volley from Jordi Alba hit the crossbar with Sevilla keeper Tomas Vaclik well-beaten.
Despite persistent Barca pressure, Sevilla won a penalty five minutes later when Oscar Mingueza crashed into Lucas Ocampos to stop a counter-attack. But Ocampos flubbed his lines from the spot, his shot easily gobbled up by Marc-Andre ter Stegen who went down and to his right.
Sevilla midfielder Fernando was shown a second yellow card in added time for pulling down Francisco Trincao at the edge of the box. Lionel Messi's free kick was deflected by the wall for a corner kick and Pique dramatically headed home from the resulting set piece to send the match to extra time.
With a man advantage, Barcelona took control of the tie five minutes into the first extra period when substitute Braithwaite's diving header of an Alba cross squirted underneath Vaclik at the goal line.
Sevilla looked to have another penalty claim moments later when a lobbed through ball into the box appeared to touch the hand of Barca defender Clement Lenglet, but a VAR review ruled in Barca's favour.
Barca get 'huge win' amid tumultuous week off the pitch
Ale Moreno is pleased to see Barcelona finally play with personality amid quite a stressful week in the tabloids.
The result means Barcelona will play the winner between Athletic Bilbao and Levante in the competition's final on April 17. The two challengers will kick off the second leg of their semi at Levante's Ciutat de Valencia on Thursday (Live on ESPN+ at 3 p.m. ET).
"I haven't been this happy for a long time, it was a magic night," said Alba.
"It's just a shame our fans weren't there to see it. We played superbly and we were always convinced we would go through but we couldn't score until the last minute. I'm so proud of the team. We have a lot of things to improve but if we fight like we fought today everything will be easier."
Barca had won 2-0 at Sevilla on Saturday to go second in La Liga, leading Koeman to declare his side had every chance of turning the tie around.
But their preparations were overshadowed when former President Josep Maria Bartomeu was arrested on Monday and the club's Camp Nou offices were raided in a probe related to allegations of improper management and business corruption.
The club will elect a new leader on Sunday and the three candidates, Joan Laporta, Victor Font and Toni Freixa, were all present in the Presidential box to witness the comeback.
Koeman described the night as his best since he took charge of the team last August and set his sights on lifting his first trophy as Barca coach after plenty of successes as a player under Johan Cruyff.
"We've taken a big step towards winning an important trophy," he said.
"We had to play a complete match and that's what we did. We kept going until the last minute, we got a bit lucky but we always believed in ourselves and we deserve the victory."
Information from Reuters was used in this report.