Eight-year-old Robert Christie and his father, also named Robert, were at a neighbour's farm on the Ballynaloob Road near Dunloy on Saturday afternoon when the incident happened.
It is understood they were found by a postman and had collapsed after being overcome by slurry fumes.
The child was airlifted to the Royal Victoria Hospital but did not survive.
The father-of-three remains in a critical condition at the Causeway Hospital.
Principal Gerry Black, from Knockahollet Primary School, where Robert attended along with his two sisters, described him as a "lovely child".
"A real open wee personality, such a caring nature," he commented.
"He lived for the weekends because that was his time to get out with his dad on the farm, and that was his passion."
Mr Black continued: "He was friends with everybody, he made time for people, he shared whatever he had with them at playtime.
"He would always say hello to you. He's going to be greatly missed by his friends, his classmates, the staff and the wider family. It's just a terrible tragedy."
Politicians, including the First Minister, deputy First Minister and Northern Ireland Secretary of State Theresa Villiers, have passed on their condolences.
This is a terrible accident and the tragic loss of a young life. The news of Robert's death will have shocked NI's farming community.
Ms Villiers said: "I send my sincere sympathies to his bereaved family and wish his father a full recovery."
The Health and Safety Executive are investigating and Chief Executive Keith Morrison has expressed his sympathy.
He said: "Incidents like this show starkly the dangers which our farming communities face and my heart goes out to those affected by this tragic accident."
Addressing the issue of farm safety, he added: "The facts tell us that farming is our most dangerous industry and that is why members of the Farm Safety Partnership - government and the farming industry together - will continue to work every day to try to avoid these tragic events occurring.'
"Work with slurry, animals, falls from heights and working with equipment continue to be the main dangers within the industry. I urge all farmers remain vigilant to the dangers and to stop and think SAFE before starting any job on the farm.
"Children on farms can be particularly vulnerable and the Health and Safety Executive will continue to work with local schools to raise awareness of the dangers on farms."
The news from Dunloy is devastating and my thoughts are first and foremost with members of the family and the local community at this terrible time.
Keith Morrison, HSENI
Neighbour and local DUP councillor Bill Kennedy said the mood was sombre in the community.
"Your heart goes out to people in a situation like that. It's devastating for everybody concerned.
"The mother is a great community worker herself, and no doubt the wider community and her friends will rally round her now in her time of need," he said.
SDLP Dunloy Councillor Harry Boyle, who knows the family, added: "They have been deeply rooted in the local community and that is clearly evidenced by the fact that the family often travelled to help elderly farmers nearby.
"Coming from a farming family, I know all too well the dangers of slurry. I share in the pain that the Christie's are experiencing right now, as will all those in the farming community."
Ballymoney Mayor John Finlay passed on his condolences on behalf of the local community.
"Robert is a farmer himself, he was helping another out and this tragedy happened, we just don't know all the circumstances but we do know a family today is in grief," he commented.
"The whole community will unite in hoping that the father pulls through and things get a little better for the family."
North Antrim TUV MLA Jim Allister said: "I think the thoughts are very much with the family, to lose a child in any circumstances is heart-rending and to lose them in the circumstances of a farm accident for this family is truly awful.
"Then to have found that the father is fighting for his life it's just an unbearable burden for them, I'm sure and our thoughts and prayers are with this family at this time."
Sinn Féin MLA for the area Daithí McKay said: "The thoughts and prayers of everybody in this rural community are with the family of those involved in this tragic incident."
The accident happened as the Ballymoney Show, a farming event, was taking place nearby.
It comes almost two years after another farming tragedy involving a slurry tank in Co Down - which claimed the lives of three family members - up-and-coming Ulster Rugby star Nevin Spence, 22, his father Noel, 58, and 30-year-old brother Graham.
Barclay Bell from the Ulster Famers' Union says work is continuing to increase safety awareness on farms.
"The work of the farm safety partnership is ongoing, we have just launched another three-year strategy and you will see farm safety remaining right at the top of our priority list as far as trying to get the message out there and trying to keep farmers thinking safe at all times," he said.
"The message, when it comes to working with slurry, is everyone should probably get out of the building, once they start to mix slurry, the message is to leave the building.
"There are masks, there are detectors available, but at the end of the day the best thing to do is vacate the house as soon as possible."