The La Liga champions entered the clash holding a 3-2 advantage from the first leg, and they moved 4-2 ahead in the tie when Raphinha netted in the 12th minute of the game at Estadi Olimpic Lluis Companys.
However, Ronald Araujo was shown a controversial red card in the 29th minute for a perceived foul on Bradley Barcola, and PSG then went on to score four times, including a double from Kylian Mbappe, to record a 4-1 win on the night and progress to the final four with a 6-4 aggregate success.
Xavi and his goalkeeping coach were also shown red cards by Kovacs in the second period, and the Barcelona head coach has been highly critical of the referee's performance.
"The referee was really bad. I told him his performance was a disaster. I don't like to talk about referees, but it had a clear impact in the season and it has to be said," Xavi told Movistar Plus.
"We are very upset and angry because the red card was the decisive factor in the match. With 11, we were in a good position, playing well and in command.
"It's too much to flash a red card in a game like this. There was another game after that... it is a pity that the work of the season was ruined by an unnecessary expulsion."
Xavi also insisted that Barcelona 'will be back', but he is still preparing to stand down as head coach of the club at the end of the 2023-24 campaign.
"We were solid in the Champions League, notable I would say. We have to learn to control small details. Next year we'll be back here. Barca will try again. I'm not important. Next year we'll be back," he added.
Barcelona now only have one trophy to fight for this season, but they are eight points behind La Liga leaders Real Madrid ahead of this weekend's El Clasico.
Ex-Barcelona forward Henry claims red card was "deserved"
Despite Xavi's views, former Barcelona forward Thierry Henry has said that the Catalan side "deserved" to be reduced to 10 men and criticised the response of the hosts.
"Barcelona deserved to go down to 10 men. That penalty was a penalty. That doesn't mean you have to concede those goals," Henry told CBS Sports.
"The goals they conceded weren't because they were playing with 10 men. It was because they started to panic. You cannot win any title or be champions when you panic. I don't care who you are.
"You cannot panic in difficult situations because it will creep up on you, it will creep up on the team in the stadium and the next thing you know, the momentum goes to Paris St-Germain. Then they find themselves out of the competition."
PSG will take on Borussia Dortmund in the semi-finals of the competition, with the German outfit beating Atletico Madrid 4-2 in the second leg of their quarter-final to secure a 5-4 aggregate win.