Hart produced an outstanding performance but to no avail as Barca wrapped up their last-16 tie by winning 1-0 at the Nou Camp to complete a 3-1 aggregate success.
Ivan Rakitic scored the only goal but Lionel Messi was at his magical best and Barca would have won more convincingly but for the brilliance of Hart.
City might have changed the outcome had Sergio Aguero taken advantage of a second-half penalty but, in truth, the scoreline flattered City.
Hart was particularly impressive after the break, when he produced a string of fine saves to frustrate Barca's much-vaunted attack, while Luis Suarez also hit the woodwork twice.
"It's disappointing - we gave ourselves a chance and didn't take it," said England international Hart.
"We've gone out to a magnificent side but that's the second time in two years which is disappointing for us.
"I had a lot to do, but I wanted it to count
I said that after the first leg, and it hasn't.
"Even when we lose - even when I make mistakes, and it's all my fault - I am still proud of my efforts, and of my team, and we will keep going.
"Make no bones about it, Barcelona were the better team in both legs, but we hung on in there, defended and made saves, and made tackles, but then didn't take our chance to get back into it.
"We have to learn to take our chances
Sometimes in Europe you won't be the better team and you have to somehow find a way to win."
The result will have done little for manager Manuel Pellegrini's long-term job prospects but this was not a night when the Chilean could be blamed.
Barcelona were simply in a different class and Pellegrini accepted as much.
He said: "It was obvious Barcelona would get more chances, we had to take risks, but Barca were at a great level with an imperious Lionel Messi.
"It is not a failure, I don't see it is a failure
It is a disappointment, but we have reached the last 16 twice.
"We were unlucky to play Barca on both occasions
We have had a big restriction on our squad size and money but we have to admit Barca were a better team.
"When Messi gets the ball, he just gets away from all the players
It creates a lot of difficulty for our players, the ball is stuck to his foot."
Despite being a clear second best, Pellegrini did think his players had their moments.
He said: "We knew what we had to do, score two goals
We took risks and fortunately for us we had Joe, who made very good saves.
"If we had scored the penalty, maybe it would not have been so quiet for Barcelona in the end."
Pellegrini has come under heavy pressure as City have faltered recently, but Hart backed him to continue.
He said: "It's always going to be the same at City, for all the players
We appreciate that if we don't win everyone will be speculated about
You have to be strong inside.
"We back our manager
He is a great manager and brought us two trophies last year
Now we have nine games to win, and that's what we hope to do."
Barcelona coach Luis Enrique felt his side should have won by a much more comfortable margin.
He said: "The score could have been a rout
Hart was incredible but we weren't really on (it) towards the end.
"We had a lot of chances, which is not normal in this type of game
We deserved to win by a bigger goal difference.
"But we achieved our objective, which was to win the game and move into the next round.
"When a team like City are down and decide to take risks, and have just three back on defence against a team like ours, we have to close out the game.
"Everything would have been much tougher if (Marc-Andre) Ter Stegen had not saved the penalty."
Source : PA
Source: PA