Francis realized he had misjudged the Chilean situation after meeting with Cruz and reading the 2,300-page report compiled by two leading Vatican investigators about the depth of Chile's scandal, which has devastated the credibility of the church in a once overwhelmingly Roman Catholic country in the pope's native Latin America.
"We have come to ask forgiveness" from the victims on behalf of Pope Francis, said Bertomeu as they arrived in Santiago.
Barros, who was a protege of Karadima, is among those accused of witnessing and ignoring his wrongdoing.
The pope accepted Barros' resignation despite previous attempts to defend the bishop.
Francis, who twice previously had not accepted Barros' offered resignation, then acknowledged that he made "serious mistakes" in judging the case due to "a lack of truthful and balanced information".
Last week it was announced they were being sent back to Chile in order to advance the process of "the process of reparation and healing of victims of abuse" in the Diocese of Osorno.
The Pope accepted the resignation of the three bishops on Monday but it was not immediately clear if the move meant that he would not accept any of the other resignations.More news: Salah remains a doubt for match against Uruguay-Egyptian FA
More news: Dancing FBI agent facing charges in accidental shooting
More news: After 'difficult' G7 summit, UK's May warns against unilateral action on trade
The highest-profile of those leaving his post was Bishop Juan Barros of Osorno, 61, who has been in the vortex of the abuse saga.
Pope Francis promised last month to Chilean Catholics scarred by a culture of clergy sexual abuse that "never again" would the church ignore them or the cover-up of abuse in their country.
"The resignation of these bishops should not dilute their criminal responsibility", he said.
Opponents have been vocal about their opposition to Barros ever since, with some of the most outspoken being victims of Karadima, who in 2011 was found guilty by the Congregation for the Doctrine of the Faith of sexually abusing several minors during the 1980s and 1990s, and sentenced to a life of prayer and solitude.
Francis angered anti-abuse activists and others in Chile when, during a trip to the South American nation in January, he told an interviewer that the accusations against Barros lacked evidence and amounted to "calumnies".
That same month he received at the Vatican three Chilean whistleblowers in the sexual abuse case.
The Catholic Church's upcoming big family rally in Ireland will feature workshops on hot-button issues facing Catholic families, including priestly sexual abuse, weathering divorce and ministering to lesbian and gay faithful.
He later apologised to victims, following criticism from Cardinal Sean O'Malley, a Vatican pointman on clerical sex abuse.