Juan Carlos Cruz, a victim of cleric Fernando Karadima in the 1980s, accused fellow priest Juan Barros of witnessing the abuse and doing nothing.
The Pope caused outrage after a visit to Chile last month by defending Bishop Barros.
The Vatican refused to comment on the letter when approached by BBC News.
Pope Francis has said in the past that dealing with abuse is vital for the Church’s credibility and perpetrators must face “sanctions”.
What allegations does the letter make?
Mr Cruz sent the text of his letter (written in Spanish) to BBC News, showing it was addressed personally to Pope Francis and dated 3 March 2015.
That was more than two weeks before the bishop’s installation in the south Chilean city of Osorno, an event dramatically disrupted by hundreds of protesters accusing Bishop Barros of covering up Karadima’s sex attacks on young boys.
The bishop has denied ever knowing about “the serious abuses” committed by Karadima, who was never prosecuted in Chile because so much time had passed but was convicted and sentenced by the Vatican to a lifetime of “penance and prayer”.
“Holy Father, I decided to write this letter to you because I’m tired of fighting, crying and suffering,” Mr Cruz writes.
“Our story is well known and there’s no point reminding you of it, except to tell you of the horror of having experienced this abuse and how I wanted to kill myself.”
In his letter, he also attaches the full text of a previous letter written a month earlier to the Vatican’s top diplomat in Chile, Archbishop Ivo Scapolo.
In that letter, Mr Cruz accuses Bishop Barros of “doing all the dirty work of Fernando Karadima”, and describes the abuse he suffered and which Bishop Barros allegedly witnessed.
“When we were in a room with Karadima and Juan Barros, if he [Barros] wasn’t kissing Karadima, he watched as one of us, one of the younger ones, was touched by Karadima and forced to give him kisses,” he writes.
“Karadima would say to me: ‘Put your mouth next to mine and stick out your tongue.’ He’d stick out his and kiss us with his tongue. Juan Barros witnessed all of this on countless occasions, not just in my case but in the case of others as well.”
Addressing himself to Pope Francis, Mr Cruz says: “Holy Father, Juan Barros says he saw nothing and yet, there are dozens of us who can testify to the fact that not only was he present when Karadima abused us, but that he, too, kissed Karadima and they touched each other.”
He concludes the letter with this appeal: “Please Holy Father, don’t be like the others. There are so many of us who despite everything think that you can do something. I treasure my faith, it’s what sustains me, but it is slipping away from me.”
Hasn’t the Pope already apologised?
Among the Pope’s remarks that caused such offence were: “The day they bring me proof against Bishop Barros, then I will speak. There is not a single piece of proof against him. Everything is slander. Is this clear?”
He also said: “No one has come forward, they haven’t provided any evidence for a judgment. This is all a bit vague, it’s something that can’t be accepted.”
The Pope later apologised for hurting victims’ feelings “without meaning to” but continued to insist there was “no evidence” against the bishop.
“In Barros’s case, it was studied,” he said. “It was restudied. And there is no evidence… I don’t have evidence to convict.”
How far did the letter get?
Members of the Pope’s own Commission for the Protection of Minors, set up in 2014 as part of efforts to counter sex abuse by clergy, have told the Associated Press they hand-delivered the letter to Francis’s top adviser on sex abuse, Cardinal Sean O’Malley, in April 2015.
Marie Collins, a member of the commission at the time, has sent the BBC a photo of her handing the letter to the cardinal.
“When we gave him the letter for the Pope, he assured us he would give it to the Pope and speak of the concerns,” she told AP earlier. “And at a later date, he assured us that that had been done.”
Mr Cruz told the BBC that Cardinal O’Malley had called him later in 2015 to say he had given the letter to the Pope.
Cardinal O’Malley, whose spokesman referred requests for comment to the Vatican, has earned respect for his work in tackling sex abuse by clergy in Boston.
Cardinal O’Malley’s disgraced predecessor, the late Cardinal Bernard Law, had moved paedophile priests between parishes rather than addressing victims’ claims.
In an unusual step, Cardinal O’Malley openly criticised the Pope last month for his initial remarks in Chile, saying he had left victims feeling abandoned.
Pope Francis announced last week he was sending the Vatican’s top expert on sexual abuse, Archbishop Charles Scicluna, to Chile to investigate the accusations against Bishop Barros.