The Premier League side lost 2-1 in the second leg of the last 16 tie at Camp Nou on Wednesday to bow out 4-1 on aggregate.
Lionel Messi's 28th goal of the season put the Spanish champions in front in the second half, but Kompany looked to have denied Barca victory on the night just a minute from time.
It was a case of deja vu for City though, as Dani Alves scored a late goal - just as he had done in the first leg - to add insult to injury for the visitors.
Despite their exit, Kompany was still full of pride for the manner in which City had approached the clash, and he is confident that they can make an impact in the competition next season.
The Belgium defender told Sky Sports: "We've got nothing left in the tank, we gave everything. We could have scored earlier which would have changed the game.
"You have to take your half chances against the best teams. Over the course of the two games there was not much between the sides. Lionel Messi is incredible player but outside of that it was even.
"Only the winners get remembered but it's only our third season in the Champions League. The way we have played means we should fear nobody next season and our time will come."
Pablo Zabaleta was sent off 12 minutes from time for a second bookable offence after showing dissent following referee Stephane Lannoy's failure to award a penalty for a challenge from Gerard Pique on Edin Dzeko - much to City's obvious dismay.
And Kompany also felt aggrieved that the French official had failed to point to the spot.
"It was a clear penalty as well. Edin Dzeko was in position to shoot and his trigger foot was caught," he added.
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