Trailing 2-0 from the first leg, Ronald Koeman's side halved the deficit early on at Camp Nou as Ousmane Dembele netted with just 12 minutes gone.
But they were forced to wait until the fourth minute of injury time to draw level, Gerard Pique coming up with the all-important goal.
Sevilla had lost Fernando Reges to a second yellow card moments earlier, and the 10 men were unable to prevent Martin Braithwaite from completing the turnaround five minutes into extra time to keep Barca's trophy hopes alive.
That is particularly significant for Lionel Messi, who is considering his future ahead of the expiry of his contract this summer.
Take a detailed look at how the Argentine performed on a thrilling evening for him and his teammates below.
stats/2308639">Barcelona v Sevilla: Player Ratings, Match Stats, Player Stats
📷 Lionel Messi: Match Snapshot
Unusually, Messi played no part in any of Barcelona's three goals as they produced a memorable comeback.
It was Dembele who got the ball rolling with a brilliant strike from distance early on, while Antoine Griezmann was responsible for the cross that Pique headed home.
And Jordi Alba set up the third with another pinpoint cross that Braithwaite met with a powerful header.
〽️ Lionel Messi: Rating Week-on-Week
Messi grabbed a goal and an assist in a 2-0 La Liga win over the same opposition last time out to earn a 9.5 FC Rating.
But, while he was clearly less effective here, the Barca No.10 still earned a match-high 8 FC Rating, which was only matched by Marc-Andre ter Stegen, who saved a penalty.
🗝️ Key Stats
Messi match stats
|FC Rating||8.0||Rank =#1|
|Shots on target||1|
|Completed dribbles||11||Rank #1|
Messi: What Next?
Barcelona now quickly turn their attentions back to a faltering league title challenge, with a trip to Osasuna awaiting on Saturday.
The Catalans must take advantage of the two Madrid clubs facing off this weekend; anything less than all three points will surely end their hopes.