Inter, who were last champions of Europe in 1965, have not been in the final of European club football's most prestigious competition since 1972.
"It's been 38 years since Inter played in the final," added Mourinho.
"We were a team of heroes, we sweated blood. I thank everyone who played and who did not play. I thank all the fans, those who were here and those who remained at home.
"Absolutely incredible things happened. Now is our festival. I want the fans at the airport waiting for these guys."
Inter went into the second leg in a strong position following their 3-1 win over Barca at the San Siro on 20 April.
But the Italians suffered an early setback at the Nou Camp when Thiago Motta was sent off on 28 minutes for putting a hand in the face of Sergio Busquets.
Busquets went down clutching his face but was caught on camera glancing up to see the ref's reaction.
Motta claimed afterwards: "He always does it, I have seen it on TV and he is holding his face and then looking at the referee - it is terrible behaviour."
The sending off looked likely to turn the tie in Barcelona's favour as Pep Guardiola's side battled to reach their second successive Champions League final and retain the trophy they won in Rome last year.
But the hosts struggled to break down a well-organised and resilient Inter, Mourinho's team producing a near-faultless defensive performance to contain their illustrious opponents.
This match was difficult with 11 players... with 10 versus 11 it was historic
Gerard Piquet eventually made Barca's pressure tell when he beat Julio Cesar in the Inter goal on 84 minutes, while Bojan Krkic had an injury-time strike disallowed after a handball by Yaya Toure.
One more goal would have taken the Catalan giants through, but Inter held on stoically for a famous triumph - and celebrated at the final whistle as if they had won the trophy.
"This match was difficult with 11 players. With 10 versus 11 it was historic," added Mourinho. "The defeat was the best of my life, but I think this team deserved to draw 0-0.
"Julio Cesar has had an amazing game, it seemed we were in equal numbers. It is an incredible joy."
Mourinho, who worked as an assistant to Sir Bobby Robson during his spell in charge at Barca, added he could never envisage being their manager now.
"I'm not stupid enough to think that this hate can be turned into love," the former Chelsea boss commented.
"I respect Barca and and I'll never forget what the club gave me in the four years I was here, but something has been created around me that is hard to make positive," he said.
"It is clear that I will end my career without having coached Barca."
Guardiola was magnanimous in defeat, choosing to emphasise the quality of Inter's defensive display after one of the few big disappointments of his career as Barca boss.
"Congratulations to Inter, they have beaten us over two games and good luck to them - any team can choose how they play and I have no complaints," he said.
Today we have experienced the cruel face of football, but now we have need to lift ourselves and think about the league
Barcelona president Joan Laporta
"We tried through the middle and down the wings, but we couldn't find a way through - it was difficult because there was no space between their defence and their midfield and they defended very well."
Barcelona president Joan Laporta urged his club to turn their attention to the defence of the Spanish league title after seeing their European dream come to an end.
The Catalan giants are a point clear of Real Madrid with four games left in La Liga but this weekend face a tough trip to improving Villarreal.
"Today we have experienced the cruel face of football, but now we have need to lift ourselves and think about the league," stated Laporta.
"We have a key game against Villarreal in order to have have some joy in La Liga. The team deserve an ovation and all our gratitude. They deserve our applause because the players have given their best at every moment.
"Now we need to recover, raise our morale and encourage the team to focus on the league. Now more than ever we need to be with this team."
For Mourinho, there is a potential treble to win with two domestic trophies to fight for in addition to the Champions League.
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