Diego Simeone's side were heading for another defeat against the LaLiga champions with 10 minutes left at the King Abdullah Stadium, but goals from Alvaro Morata and Angel Correa turned the semi-final tie around.
The 3-2 victory was Atletico's first win over Barcelona since April 13, 2016 and set up a derby showdown against Real Madrid on Sunday, between the two invited teams for this new-look tournament .
Barca boss Ernesto Valverde would have been frustrated with his team's late capitulation after goals by Lionel Messi and Antoine Griezmann put them on course for the final.
"We played a tough game, we started well in the first 10 minutes and then they were superior," said Simeone, in quotes reported by Mundo Deportivo.
"We always have the plan of trying to play our game, but the opponent is very good. Messi had one more step. When he had the ball it seemed that something was going to happen.
"We talked at half-time and we started with that goal that gave us strength, but they quickly got level with us.
"The two goals came and two disallowed (for Barca), and with it a disturbing offensive game rhythm. Changing (Marcos) Llorente and the position of Correa up front gave us more options.
"And in the 76th minute another team was seen, with strong legs, which showed that we could win."
Simeone is now preparing his team to take on Real, with Zinedine Zidane's team having had 24 hours more rest than Atletico after their 3-1 success over Valencia on Wednesday.
But the Argentinian commented: "We are going to play a final against a team that made a fantastic match, even with the absence of (Gareth) Bale and (Karim) Benzema.
"They responded in a very good way, with a very nice game. Madrid has mobility, speed and precision. They play to win. Sunday will be a tough game."
VAR was prominent throughout, with Messi having one goal ruled out for handball and Gerard Pique denied after Arturo Vidal was deemed offside.
Atletico also felt they should have had a penalty and despite the technology looking at a potential handball from Pique, nothing was given.
Barca forward Griezmann was quoted by AS as saying: "We have VAR to help football. Sometimes it helps you and sometimes it goes against you, but it is what it is."
The former Atletico star added: "In the same way I can't explain our mistakes, and in the end they had more energy and it cost us. We have to watch the game back, work and improve.
"I failed to make the pass to (Samuel) Umtiti. Little things like that can result in you losing a match, a league, a Champions League. We will work and that's it."
