He also promised to hold bishops accountable for the protection of minors.
Victims’ advocacy groups around the world have pressed the Vatican for decades to severely discipline any complicit church hierarchy.
While the Pope spent his morning with the three men and three women, listening to their stories one by one, several victims’ groups blasted the meetings as being “a PR event”.
The Vatican said Pope Francis expressed personal “sorrow” in his homily at the private Mass with the victims for the “sins and grave crimes” of clerical sex abuse against them.
He also said: “All bishops must carry out their pastoral ministry with utmost care in order to help foster the protection of minors, and they will be held accountable.
“I beg your forgiveness, too, for the sins of omission on the part of Church leaders who did not respond adequately to reports of abuse made by family members, as well as by abuse victims themselves.
“This led to even greater suffering on the part of those who were abused and it endangered other minors who were at risk.
“Before God and his people, I express my sorrow for the sins and grave crimes of clerical sexual abuse committed against you. And I humbly ask forgiveness.”
Vatican spokesman the Rev Federico Lombardi said two Irish, two British and two German victims met the Pope.
The six adults who had been sexually abused as youths by clergy met privately for about a half an hour each with Pope Francis at the Vatican.