Lionel Messi and Dani Alves both scored again as Barca advanced to the quarter-finals with a 2-1 win in the second leg at the Nou Camp to progress 4-1 on aggregate.
Just as in the first leg, when Martin Demichelis was dismissed, Manuel Pellegrini's team were reduced to 10 men thanks to a red card for Pablo Zabaleta.
But City goalscorer Kompany felt his players could walk away with their heads held high, having put in a spirited display - in 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 on Sky Sports 1: "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."