Juan Carlos Cruz, James Hamilton and Jose Andres Murillo spoke to reporters on Wednesday after spending five days with the pope at his Vatican hotel.
Their press conference was broadcast live in Chile, a sign of the unprecedented nature of their hours of meetings with the pope.
Mr Cruz said that during his private encounter with Francis, the pope acknowledged: "I was part of the problem.
"I caused this, and I apologise to you."
"I believe that he was sincere," Mr Cruz said.
Mr Cruz said he believed that Francis was simply misinformed about the case of Bishop Juan Barros, whom the three men have long accused of having witnessed and ignored their abuse.
Mr Barros was a protege of the Reverend Fernando Karadima, a charismatic preacher and darling of Chile's conservative Catholic society who was removed from ministry and sentenced by the Vatican in 2011 to live in penance and prayer for having sexually abused minors.
Mr Barros and other Karadima-trained bishops never acknowledged having witnessed his abuse, even though his victims have long placed them at the scene.
Francis had strongly defended Mr Barros during his January trip to Chile, calling the accusations against him "calumny".
He claimed to have never heard from victims about Mr Barros, even though he had received in 2015 a letter from Mr Cruz detailing Mr Barros' wrongdoing.
Mr Cruz had written to Francis after the pope overruled opposition from some Chilean bishops and appointed him bishop of Osorno.
Just this week, a former Chilean minister revealed that the Chilean government too had wanted Mr Barros out as the preacher to the Chilean armed forces, and was "surprised" when Francis named him instead to head up Osorno.
Mr Cruz said he did not press the pope on what he knew or when.
But he said he warned him about the "toxicity" of the churchmen who had "duped him," naming the current and former archbishops of Santiago, the Vatican's ambassador to Chile, and members of the Chilean bishops' conference.
Mr Hamilton said they will probably never know the full truth about what the pope knew, but that the important thing is that the pope now is "very well-informed".
He said he was prepared to wait to see what concrete action he will take.
"Everybody deserves, especially in this case, a second chance," Mr Hamilton said of the pope.
The three men did not say what exactly they want Francis to do.
But previously, they have called for Mr Barros and other Karadima-trained bishops to resign, as well as a handful of other Chilean bishops with poor records on dealing with abuse cases.
Francis has summoned the entire Chilean bishops conference to Rome later this month for a dressing down and to plot reforms in the church.
In a statement, the men said they saw the "friendly face" of the church this week, after having been treated for 10 years as "enemies" of the Chilean hierarchy.
But they warned that unless Francis takes concrete action, their talks will be in vain.
They said abuse and cover-up are not just sins "but crimes and corruption that do not end in Chile but are an epidemic".
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