Irish survivor Marie Kane, who was among the victims who met with Francis, said later she believed the pope spoke from the heart but warned if the Catholic Church did not change it would "disappear."
In his homily delivered Monday in his native Spanish, Francis said he felt "deep pain and suffering" for the "despicable actions" of priests and bishops who had abused their vocation and violated their victims.
"I ask for the grace to weep, the grace for the church to weep and make reparation for her sons and daughters who betrayed their mission, who abused innocent persons," he said during the Mass held at the chapel next to the Vatican residence where he lives.
"All bishops must carry out their pastoral ministry with the utmost care in order to help foster the protection of minors, and they will be held accountable," he said without elaborating on what accountability might look like.
Francis noted that abuse was often a source of "unrelenting emotional and spiritual pain, and even despair," and expressed his " love and sorrow" for families who had also suffered from the suicide of a victim.
"I beg your forgiveness, too," he said, "for the sins of omission on the part of church leaders who did not respond adequately to reports of abuse."
Kane, a 43-year-old mother of two, was abused as a teenager. Speaking publicly for the first time, she said the cover-ups in the church had been "unbelievable," and she hoped Francis could stop the practice of shuffling abusive clerics from one parish to another.
"He said he was sorry for what happened; I felt it came from the heart," Kane told RTE Radio. "I think he realizes he has a lot of changes to make; I would like to believe he will. The church will disappear if it doesn't change."
Francis was accompanied by Boston Cardinal Sean O'Malley, one of his closest advisers and the man tapped to help clean up the massive abuse scandal in Boston a decade ago. O'Malley also heads the pope's blue-ribbon task force to make suggestions on implementing abuse reforms around the world.
But the Survivors Network of those Abused by Priests, which represents 18,000 members around the world, said far more needed to be done to stop what it called an ongoing crisis in the church.
A Pew Research Center survey released last December showed that 70 percent of U.S. Catholics want the abuse scandal to be a "top priority" for the pope, and the issue ranked as "the most important" problem facing the church.
"Children are being assaulted by clerics right now. Bishops are concealing these crimes right now," SNAP President Barbara Blaine said in a statement. "And Francis must take decisive action right now, action to expose and remove clerics who commit and conceal heinous crimes against the most vulnerable."
Another U.S.-based group, Bishop Accountability.org, said the pope had been "largely silent" on clerical abuse in his native Argentina and he must remove and discipline abusive clerics.
The Vatican's chief spokesman, the Rev. Federico Lombardi, rejected suggestions that Monday's meeting was simply a public relations exercise.
Asked why the meeting included no American victims, even though the U.S. scandal prompted worldwide attention, Lombardi said it was "not such an easy thing to organize an event like this" and the intention was to create a moving experience.