Spence, 22, who was recently named in the Ireland international squad, was killed along with his brother and father on Saturday evening.
A fourth person, understood to be their sister, also fell into the tank and was taken to hospital where she was recovering from the effects of inhaling fumes.
The accident happened at a farm in Hillsborough, County Down, about ten miles from Belfast.
Investigations into the cause of the tragedy are ongoing but there are reports that family members may have been attempting to rescue a dog from the tank, before Nevin helped in the effort to save his relatives.
The Health and Safety Executive is to investigate the circumstances and establish a clearer picture of how the tragedy unfolded.
Emergency services were called to the site shortly after 6pm last night.
Firemen pulled all four victims from the slurry pit.
Two of the men were declared dead at the scene, while the third was taken to hospital, where he died shortly afterwards.
The site is only nine miles from another farm, where a farmer died in June last year after falling into a slurry tank.
The large tanks are used to store slurry – a combination of cow manure and water – until it is ready to be used as fertilizer.
A spokesman for the Northern Ireland Ambulance Service said: “One of the four showed initial signs of responding to the efforts of the crews and was transported immediately to the nearby Lagan Valley Hospital, where unfortunately he died a short time later.
“Despite the best efforts of all involved the other two men died at the scene. A female was treated at the scene and taken to the Royal Victoria Hospital where she is recovering from the effects of fume inhalation.”
Pat Catney, a Social Democratic and Labour Party councillor, said: “I’m shocked and devastated. It’s another farming accident, and there are too many of them.
“Something has to be done to stop these accidents. I’m sorry for the family and for the neighbours,” he told UTV.
The Health and Safety Executive has been notified and will carry out an investigation.
Spence’s club, Ulster Rugby, confirmed he was one of the three victims.
He played for Ulster Ravens, their second team, against Munster A on Friday afternoon.
Spence was considered a rising star of rugby union, having represented Ireland at under 20 level and being called to the Ireland training squad for the Six Nations last year.
In May this year, he was selected for Ireland squad ahead of a match against the Barbarians, a team made up of players from across the world.
A former pupil at Wallace High School, Spence made his first appearance for Ulster against Ospreys in April 2010.
Rugby players and fans paid tribute to the young player.
Northern Ireland First Minister Peter Robinson said on Twitter “Deeply saddened to hear of the tragedy in Hillsborough this evening. My thoughts and prayers are with family at this time of devastation.”
The Ulster Farmers’ Union said the tragedy had shocked the entire farming community.
UFU president Harry Sinclair said: “On behalf of all our members, I would like to extend our deepest sympathies to the family involved.
“This news has shocked the farming community. Our thoughts are with the family and friends of the three men who have so tragically lost their lives.
“We understand that a fourth member of the family has been hospitalised and we pray that they will make a full recovery.”
Nevin Spence’s management company expressed their “deep regret” after the Ulster centre’s death in a farming accident.
Cornerflag Management, who represented Nevin Spence, released a statement saying: “It was with great sadness and deep regret that we learned of the sudden and tragic passing of Nevin, his father Noel and brother Graham in an accident on their family farm on Saturday afternoon.
“Our thoughts and prayers are with the Spence family at this extremely sad time.”
Welsh rugby legend Shane Williams said, “My thoughts and prayers go out to the family of Nevin Spence. A tragic loss. A class act on the field also. RIP”. New Zealand rugby world cup winner Dan Carter tweeted: “Such sad news and my thoughts go out to his family”.
Northern Ireland golfer Rory McIlroy also tweeted his condolences, saying: “Just heard the tragic news of Nevin Spence and his family. Makes you cherish every day you have on this Earth”.