The Spanish giants went into the second leg leading 3-2 on aggregate and got off to a perfect start when Raphinha poked them in front.
2
2
But the tie was turned on its head when defender Ronald Araujo was sent off for a last-man foul on Bradley Barcola.
Former Barcelona star Ousmane Dembele equalised before half-time and PSG levelled the tie after the break through Vitinha’s long-range effort.
Kylian Mbappe then scored twice to give his side a 6-4 aggregate win and set up a semi-final clash against Borussia Dortmund.
And the result means Barcelona will miss out on the expanded Club World Cup next year.
The competition will include 12 teams from Europe in 2025 rather than the usual one, and will involve the Champions League winners between 2021-2024.
The Uefa club coefficients decide the other eight places, although seventh-placed Barca are still set to miss out.
That is because they rank lower than Atletico Madrid, who will enter the Club World Cup along with 2022 Champions League winners Real Madrid.
And since no more than two clubs from the same country can qualify, there will be no place for Barcelona.
Most read in Champions League
BEST FREE BET SIGN UP OFFERS FOR UK BOOKMAKER
Fans claimed Barca were robbed by the referee as manager Xavi also hit out over Araujo’s red card.
Xavi himself was sent off during the match and afterwards he raged: “It’s a pity, our Champions League is over due to the referee’s mistake.
“I just told the referee that he’s been a disaster. It’s the reality.”