The error was made by a police officer at Christchurch International Airport on February 7 last year.
The father of the child issued civil proceedings on his own behalf and for his child in the High Court which were to be heard on August 3.
Police say those civil proceedings have now been settled "following an agreement being reached between the parties".
"Police unreservedly apologise for the error that was made, and for the substantial distress caused to the plaintiffs," a police spokesman said.
"Police are totally committed to ensuring that we deliver the services that the community expect and deserve."
Airport police "deal with these orders routinely" in New Zealand's international airports, the spokesman said.
Police policy is that once a border alert confirming a Family Court order is activated, the affected passenger will not be permitted to travel in the absence of proof that the order has been discharged, or varied to allow travel.
"A full investigation was conducted as to why that did not happen in this case and improvements have been made to airport processes and practices to ensure it does not happen again," the police spokesman added.