The three men drowned in the slurry tank on their family farm in Hillsborough, County Down, on Saturday evening.
Family members have since explained that father Noel fell into the tank while trying to save his pet dog, with his two sons killed while trying to rescue their dad.
All three were overcome by toxic fumes in the incident, with sister Emma - who was present at the accident - also suffering from fume inhalation.
Emma Spence has since been released from hospital.
Grieving family members said in a statement released by their church: "The families of Noel, Graham and Nevin Spence are trying very hard to come to terms with their tragic loss.
"The three men were very close to each other in life, and that love was expressed in their final moments trying to help one another.
"The family is being supported and comforted by other family members, friends and neighbours.
"Arrangements for a thanksgiving service for the three is currently under way and details will appear in the press in due course."
The 22-year-old Nevin, who played centre for Irish province Ulster, played an exhibition game for the Irish senior team against the Barbarians this year, but had not made his full competitive debut.
He was widely expected to become a full Irish international in the future.
Graham, 30, was a father of two.
A slurry tank contains animal waste and other biodegradable material that is turned into organic fertiliser.
"Nevin was an outstanding young player," Irish Rugby Football Union President Billy Glynn said in a statement. "The game has been robbed of a very talented young man."
Spence, who was named Young Player of the Year at the Irish Rugby Football Union Players' Association Awards in 2011, played three games for the second-tier Ireland Wolfhounds.
He had an injury-blighted start to the current season and played for the Ulster Ravens against Munster A on Friday as he worked his way back to full fitness.
New Zealand fly-half Daniel Carter, golf's world number one Rory McIlroy and Wales winger George North were among those who expressed their condolences on Twitter.
"Devastating news about Nevin Spence, his dad and brother. What a wonderful talent he was," said Ireland captain Brian O'Driscoll.