The Blaugrana came into this clash on a two-goal deficit, having lost the first leg of their semi-final tie at the Estadio Sanchez Pizjuan in February. However, they would have been buoyed by their 2-0 league victory against the same opponents on Saturday.
Barcelona manager Ronald Koeman named an unchanged line-up from the side that picked up that win in Andalusia, with all 11 players retaining their positions in a 3-1-4-2 formation.
Just like on Saturday, it was Dembele who opened the scoring. The France international had seen an attempted one-two with Lionel Messi intercepted. He then showed great skill and strength to hold off the challenges of three Sevilla defenders before finding the top corner with a stunning shot from 22 yards.
Both sides traded chances after this, with close calls recorded at both ends. It was, however, Barcelona who came closest to doubling their lead, with Lionel Messi's goal-bound shot cleared off the line, while Pedri shot wide from close range.
Nevertheless, neither team was able to find the back of the net to ensure that Barcelona went into the break one goal ahead.
Aleix Vidal suffered a hamstring injury at the start of the second half and had to be substituted soon after the ineffectual Luuk de Jong had also departed.
Jordi Alba saw a bicycle kick cannon off the crossbar after some excellent work by Ousmane Dembele.
Sevilla were awarded a penalty right after that, with referee Jose Maria Martinez pointing to the spot after substitute Lucas Ocampos was felled by Oscar Mingueza in the 70th minute.
Lionel Messi protested the decision heavily but the referee was having none of it and issued a yellow card to the Barcelona captain.
Ocampos dusted himself up to take the spot-kick but his effort was comfortably saved by Marc-Andre Ter Stegen. In all honesty, it was a terrible spot-kick by the Argentine, and his expression after the penalty said it all.
The final 15 minutes saw Barcelona asking all the questions. However, a typically resolute Sevilla defense kept them at bay.
The visitors were reduced to 10 men in the second minute of injury time after Fernando was shown a second booking for fouling Francisco Trincao.
Barcelona scored their second goal soon after. Lionel Messi sent in a corner kick that was not properly dealt with, and Diego Carlos inexplicably headed the ball into Griezmann's path rather than allow it to go out for another corner kick.
The France international did not need a second invitation to send a cross into the box. Gerard Pique then rose highest to force extra-time with virtually the last kick of the game.
Barcelona kept their momentum going in the extra 30 minutes and went ahead just five minutes into extra-time.
Jordi Alba sent a cross into the area, and Braithwaite attacked it with a header that went through the legs of Thomas Vaclik.
This gave the Catalans the advantage for the first time in the tie, and it was one they did not let go.
There were wild appeals from the visitors for a handball by Clement Lenglet but the VAR waved play on after a lengthy review. Gerard Pique also caused concern when he hobbled off with an apparent knee injury, but the World Cup winner finished the game.
The victory sees Barcelona qualify for the final of the Copa del Rey where they will take on one of either Athletic Bilbao or Levante.
Here are five talking points from the highly entertaining fixture.
#5 Lucas Ocampos' poor penalty costs Sevilla
Lucas Ocampos started the game on the bench for Sevilla but was introduced in place of Oliver Torres in the 62nd minute.
The Argentine winger is one of the most dynamic and influential players for Julen Lopetegui, and it took less than eight minutes for him to make a mark on the game.
Having won the penalty, the 26-year-old stepped up to take the spot-kick. Considering his pristine record from the spot this season, it was expected that he would at least test the Barcelona goalkeeper.
However, the former Monaco man produced a poor effort that was comfortably saved by Ter Stegen.
Had he scored, Sevilla would have reclaimed the initiative in the tie with a crucial away goal that would have all but guaranteed their spot in the final.
Unfortunately for the Andalusians, that failed to happen and they ultimately paid the price for their number 5's frugality.
#4 Barcelona keep title hopes alive
Barcelona have been poor throughout this season and currently face an uphill task to reclaim La Liga.
Their 4-1 trouncing by Paris Saint-Germain in the UEFA Champions League also means that they are on the verge of elimination on the continent.
They came into this Copa del Rey second leg fixture on a two-goal deficit, and few would have given Ronald Koeman's side any chance of overturning it.
However, a comprehensive display, coupled with Pique's last-gasp strike, saw them live to fight another day in the competition.
The result drew shades of their quarter-final victory over Granada, where two late goals forced extra-time.
Having ended last season trophyless, it would have represented a major travesty for Barcelona to go two campaigns without a trophy. However, the Catalans turned the tie around to ensure that their quest for a record-extending 31st Copa del Rey title goes on.