Mandela's death aged 95 at his home on Thursday sparked a wave of emotional tributes from sportsmen from South Africa and across the world.
SARU chief Oregan Hoskins said in a statement: "All of our lives are poorer today at the extinguishing of the great beacon of light and hope that led the way for our country through the transition to democracy.
"His name will rank among that of the greatest liberators and humanitarians for as long as mankind walks the earth. It was our privilege to have lived in this country during his lifetime.
"Madiba was a true icon of inspiration and as much as South Africa owes so much to him, so does rugby.
"Through his extraordinarily vision, he was able to use the 1995 Rugby World Cup as an instrument to help promote nation building just one year after South Africa's historic first democratic election."
Mandela unforgettably presented South Africa's rugby union captain Francois Pienaar with the World Cup in 1995 at Ellis Park, Johannesburg. It was the first major sporting event to be staged in South Africa after the 1994 democratic elections.
The image of Mandela, dressed in a Springbok rugby shirt and cap, handing over the Webb Ellis trophy has become one of the most iconic and evocative images in the history of sport.
Hoskins added: "Mr Mandela was also instrumental in retaining the Springbok as the emblem for our national team at a time when a chorus of voices advocated a change of the symbol, for various reasons. It was an act of reconciliation and generosity of spirit which no one could have expected.
"The South African Rugby Union also sincerely appreciated the many times he inspired the Springboks to many great heights on the playing field. However, Mr Mandela inspired the entire South African nation with his kindness, generosity and ability not only to forgive, but also to understand his fellow citizens."
Springbok captain Jean de Villiers said: "My lasting memory of Madiba is that of a person who had enormous ability to bring people together. His presence at a Test Match just lifted the crowd and energised the team - it is actually hard to describe.
"Of course, as a sportsman I am so grateful for him for what he did for our country. He inspired South Africans, who for so long were very divided, to peacefully build a united Rainbow Nation."
Joost van der Westhuizen, who played in the 1995 World Cup final, said on Twitter: "A sad day for our country. Rest in Peace Madiba. Condolences to his family and friends."
Current South Africa international Bryan Habana wrote: "Thank you for the inspiration and hope. May your legacy live on forever. #forevergrateful #proudlySouthAfrican."
John Smit, the most-capped Springbok in history, tweeted: "A day we knew was coming but hoped never would, a man we didn't have for long enough but left a lifelong legacy. Madiba we will never forget."
Cricket South Africa said on their official Twitter account: "RIP Tata Mandela. It is because of you that a represented Proteas team can express their talent across the globe #mandela."
South Africa's one-day international captain AB de Villiers saluted the legacy of Mandela, tweeting: "Let us now, more than ever, stick together as a nation! We owe him that much. #madiba you will be missed! #tata #inspiration #leader."
England and Australia Cricket confirmed both teams would wear black armbands and observe a minute's silence ahead of the second day of the second Ashes Test.
Giles Clarke, the England and Wales Cricket Board chairman, said: "Nelson Mandela was a truly inspirational statesman for many generations of South Africans as well as many peoples around the world. Under his patronage South African sports, including cricket, emerged from the dark shadow of apartheid.
"In the past 25 years South African cricket has prospered in Nelson Mandela's 'Rainbow Nation'. There is no doubt that Nelson Mandela's positive influence on not only South Africa but also the world in general has contributed to that sporting prosperity."
FIFA president Sepp Blatter paid an emotional tribute to his "dear friend", hailing the 95-year-old's impact on the 2010 World Cup in South Africa.
He said there would be a minute's silence as a mark of respect ahead of the next round of international matches
Blatter said: "It is in deep mourning that I pay my respects to an extraordinary person, probably one of the greatest humanists of our time and a dear friend of mine: Nelson Rolihlahla Mandela.
"He and I shared an unwavering belief in the extraordinary power of football to unite people in peace and friendship, and to teach basic social and educational values as a school of life.
"When he was honoured and cheered by the crowd at Johannesburg's Soccer City stadium on 11 July 2010, it was as a man of the people, a man of their hearts, and it was one of the most moving moments I have ever experienced. For him, the World Cup in South Africa truly was 'a dream come true'.
"Nelson Mandela will stay in our hearts forever. The memories of his remarkable fight against oppression, his incredible charisma and his positive values will live on in us and with us.
"As a mark of respect and mourning, the flags of the 209 member associations at the home of FIFA will be flown at half-mast and there will be a minute's silence before the next round of international matches."
Pele called Mandela "one of the most influential people in my life" and "my hero" as he offered his condolences on Twitter.
The Football Association said on Twitter: "The Football Association is sad to hear about the death of Nelson Mandela. May he rest in peace."
Usain Bolt hailed Mandela as "the world's greatest fighter".