Barca's 3-1 home win over Napoli saw them advance 4-2 on aggregate in the last-16 tie. The five-time Champions League winners had failed to progress from the group stage in the last two seasons and were defeated in the round of 16 in 2020-21.
Barcelona, who are third in LaLiga and trail leaders Real Madrid by eight points, have struggled for consistency this season and Xavi announced in January that he would leave the club at the end of the season.
"We have received unfair criticism," Xavi told reporters. "There has been too much pressure on the players like ultimatums, life or death.
"I even read that we were the joke of the Champions League. And now what?"
Xavi had said before the match that the second leg of the last-16 tie was Barca's biggest game of the season.
"We had to be better than Napoli and we were, it's a more than deserved qualification, it's one of my best moments as Barcelona coach.
"We can be proud because we are one of the best eight teams in Europe."
