In a surprise decision on Thursday, the Lyon court handed Barbarin a six-month suspended prison sentence for not reporting the facts in the period between July 2014 and June 2015.
Rev Bernard Preynat's alleged victims said Barbarin and other church officials covered up for him for years, but the statute of limitations had expired on some charges and even the victims had expected that the cardinal would be acquitted.
The prosecutor had also argued against convicting, saying there were no grounds to prove legal wrongdoing.
Preynat is accused of abusing Boy Scouts in the 1970s and 80s and will be tried separately.
Following the verdict, Barbarin said he would soon offer his resignation to the Pope.
Barbarin was not in court for the decision, but in a brief statement afterwards he said: "I have decided to go and see the Holy Father to offer him my resignation."
He spoke of his "compassion" for the alleged victims and said they are in his prayers.
His lawyer Jean-Felix Luciani said he will appeal against the verdict, which he said was "not fair at the juridical level".
Speaking of his "disappointment" at the sentence, Mr Luciani added: "We hope that at the next step, justice will be done" - suggesting he is hopeful of acquittal in the appeal.
Five other defendants were acquitted.
Alleged victims of Preynat said Barbarin's conviction is a victory for child protection.
Francois Devaux, president of the victims' association La Parole Liberee (Lift the Burden of Silence), said: "We see that no-one is above the law. We have been heard by the court. This is the end of a long path.
"This is a victory that sends a strong signal to lots of victims and a signal to the church as well."
A lawyer for some of Preynat's alleged victims, Yves Sauvayre, called the verdict "historic".
He added: "This is a turning point."
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