The City captain promised his team had given their all and said they would return to the Champions League next season not fearing anyone.
Lionel Messi and Dani Alves were again the executioners as Barcelona eased into the quarter-finals. Just as in the first leg, when Martin Demichelis was dismissed, Manuel Pellegrini's team were reduced to 10 men after a red card for Pablo Zabaleta.
But Kompany, who scored City's goal, felt his players could walk away with their heads held high, having put in a spirited display, in marked contrast to their weekend FA Cup exit at the hands of Wigan.
Asked if it would be hard for City to lift themselves ahead of their return to Barclays Premier League action, the Belgium defender said: "No. If anything, it should give us motivation. The way we played against Wigan, the performance today was completely different.
"Only the winners get remembered, but it's only the third year ever for Manchester City in the Champions League. I'm looking at this and I don't see why next year we should fear anyone."
Edin Dzeko charged into the area and had strong claims for a penalty as he went down under a Gerard Pique challenge.
City's players were furious nothing was given, but Zabaleta took his protests too far and was given his marching orders with a second yellow card 12 minutes from time.
"For me, it was a clear penalty," Kompany said. "He is in a position to shoot and his triggering foot is getting caught - it takes the chance away. But we can discuss stuff like this - you never know whether it would have put us in a position to qualify."
He added: "I think I speak for the team in saying we've got nothing left in the tank. We gave everything. A few times we could have scored, and obviously that would have changed the game. Going for a second time down to 10 men after it was a clear penalty, it's not easy conditions to come and play against a strong team.
"Over the course of two games, it's funny I say this, but whenever there was 11 players [against 11] there wasn't much between the two teams.
"Obviously, Messi is an incredible player and whenever he has the ball danger can happen - but outside of that we had huge changes. If we score, we're looking at a completely different game."
Barcelona looked far from the classic team who lifted the trophy in 2009 and 2011 but a stuttering version under Gerardo Martino that had lost two of their last three games was still far too strong for City.
Messi put his side in front by producing a neat finish with the outside of his left boot from close range in the 67th minute after defender Joleon Lescott failed to stop a routine through-ball from Cesc Fabregas.
"We knew we had to score a goal to bring some calm to the tie, as we were aware of how dangerous City are," said Messi. "I don't think the tie was ever in danger for us, and we controlled the match very well. I think tonight we got back to being the Barcelona that we all want to see and we were able to beat a very tough opponent.
"I don't know if the criticism [in the media after the La Liga defeat to Valladolid] was over the top. We are the first to criticise ourselves and we didn't play the way we should have done there.
"We don't let the criticism affect us and we just focus on ourselves and try to turn things around and that's what we did tonight."