The family filed the civil suit in federal court in 2011 against Finn, the diocese of Kansas City-St. Joseph, Mo., and Fr. Shawn Ratigan, who pleaded guilty last year to charges of producing child pornography.
Ratigan had taken hundreds of lewd and suggestive photos of young children; the lawsuit, which was settled Tuesday, was filed by the parents of a girl who was 2 years old when Ratigan started photographing her in 2006.
"We hope this settlement comforts at least some of the many families who have suffered and are suffering because Bishop Robert Finn refused to call police, protect kids and monitor Father Shawn Ratigan," said Barbara Dorris of SNAP, the Survivors Network of those Abused by Priests.
In September, in an arrangement with prosecutors, Finn was convicted of a misdemeanor charge of failing to report Ratigan to authorities as required by law.
Finn and the diocese received multiple complaints about Ratigan in the months preceding the priest's arrest in 2011 and church officials had seen some of the pictures of children on his laptop computer in December 2010.
Despite calls for Finn to resign, he has remained as head of the diocese.
Diocesan spokesman Jack Smith told The Kansas City Star that the payout would be covered by insurance.
The tab for Finn's own defense amounted to $1.4 million, the diocese said last year, which was covered by insurance and funds collected from parishes.
The judge in the civil lawsuit dismissed the claim that Finn and the diocese aided and abetted Ratigan in possessing pornography.
The diocese previously paid out $10 million in 2008 to settle cases by plaintiffs who alleged sexual abuse by 12 priests.
It still faces dozens of other lawsuits related to abuse allegations against other priests.