PSG had won the first leg 4-0 in France and, despite netting three within 50 minutes at Camp Nou, Edinson Cavani's away goal meant that Barca needed to score a further three times to go through.
However, Neymar notched with two minutes remaining, added another from the penalty spot in stoppage time and then laid on the clinching goal for Sergi Roberto to secure a remarkable 6-5 aggregate victory.
Lionel Messi had earlier scored a spot-kick, while Javier Mascherano avoided punishment for an apparent handball in his own area, and PSG coach Emery was unhappy that the officials sided with Barca on more than one occasion.
"We have only shown our personality at times, but the refereeing with two penalties for them and the penalty not given to us has had an impact," he said.
"And then Barcelona also had the crowd dragging them home. This makes the most in-depth analysis difficult. We believed we could do it, but the referee's decisions penalised us.
"We had a clear opportunity with [Angel] Di Maria and another with Cavani, but, in [the last] five minutes, in this incredible game, everything happened.
"When we analyse it it is true that we have missed a major opportunity. The first half was our fault and we didn't manage to press or do things with the ball. We defended very deep and they had more chances to score. We made mistakes for the two first-half goals.
"We were calmer in the second half and better positioned. We improved and had a chance for 3-2. The refereeing decisions went against us and we lost everything in the closing minutes. Barcelona are capable of doing that. It was all or nothing for them in the final minutes.
"It's a negative experience for me and for the club. We need to learn from it."