Barcelona could not hold on to Ousmane Dembele's first half lead and went on to draw 3-3 with the Serie A side at the Camp Nou.
Robert Lewandowski scored a late brace but Nicolo Barella, Lautaro Martinez and Robin Gosens scored for Inter to give Simone Inzaghi's side the best chance of joining Bayern Munich in the knockout rounds.
Following a heavy outlay during the summer transfer window with the signings of Robert Lewandowski, Raphinha, Jules Kounde, Andreas Christensen, Marcos Alonso and Franck Kessie, big things were expected of Barcelona in Europe but instead they face demotion to the Europa League for a second straight season.
Xavi could not hide his disappointment in the performance of his players and took full responsibility.
"If you don't beat Inter at home you don't deserve to go on," the Blaugrana head coach told reporters, with Inter needing just one win from their final two games to progress and eliminate Barca.
"Now it doesn't depend on us anymore, we don't deserve to continue in the Champions League. It is a hard blow, very hard. The word would be cruel.
"If one player or the other makes a mistake as a manager, it's my responsibility. I take full responsibility when my players make mistakes.
"We had to stay focused and instead we went on the pitch badly in the second half, but it remains a mistake of the coach.
"In front of such extraordinary supporters, you had to do more. I'm disappointed and this competition is cruel to us."
Barcelona are back in action against Real Madrid in La Liga this weekend.
