Expedition director Mensun Bound described the Endurance as "by far the finest wooden shipwreck I have ever seen".
"It is upright, well proud of the seabed, intact, and in a brilliant state of preservation."
He said the find was "a milestone in polar history" and would help bring Shackleton's story to the next generation.
"We hope our discovery will engage young people and inspire them with the pioneering spirit, courage and fortitude of those who sailed Endurance to Antarctica."
Expedition leader Dr John Shears described the mission as "the world's most challenging shipwreck search".
He said the moment cameras caught the ship's name was "jaw-dropping".
"In addition, we have undertaken important scientific research in a part of the world that directly affects the global climate and environment.
"We have also conducted an unprecedented educational outreach programme, with live broadcasting from onboard, allowing new generations from around the world to engage with Endurance22 and become inspired by the amazing stories of polar exploration, and what human beings can achieve and the obstacles they can overcome when they work together."
Another of those involved, Subsea project manager Nico Vincent said state-of-the-art underwater technology was used to find the wreck.
The team that found it sailed on a South African polar research and logistics vessel, S.A. Agulhas II.
Trust chairman Donald Lamont said its outstanding master and crew combined with a committed expedition team containing members from the UK, South Africa, Germany, France and the US had succeeded in "this historic achievement".
The Imperial Trans-Antarctic Expedition
It was Sir Ernest Shackleton's ambition to achieve the first land crossing of Antarctica from the Weddell Sea via the South Pole to the Ross Sea.
The Ross Sea Party which was landed at Hut Point on Ross Island had the task of laying supply dumps for Shackleton's crossing party, and achieved its objective, but at the cost of three lives lost.
In the Weddell Sea, Endurance never reached land and became trapped in the dense pack ice and the 28 men on board eventually had no choice but to abandon ship. After months spent in makeshift camps on the ice floes drifting northwards, the party took to the lifeboats to reach the inhospitable, uninhabited Elephant Island.
Shackleton and five others then made a 1300km journey in the lifeboat, James Caird, to reach South Georgia. Shackleton and two others then crossed the mountainous island to the whaling station at Stromness.
From there, he was able to mount a rescue of the men waiting on Elephant Island and bring them home without loss of life.
