The Victorian-era vessel became part of nautical folklore after it vanished in the mid-19th Century.
Its captain, Sir John Franklin, had been searching for the fabled Northwest Passage.
Experts on Thursday confirmed that the wreck, discovered last month, was indeed the celebrated Royal Navy vessel.
"It is in astonishing condition,'' said search team member John Geiger, president of the Royal Canadian Geographical Society. "We're over the moon."
The ship set sail from England in 1845 under Sir John's command.
He was accompanied by a second ship, HMS Terror, captained by Francis Crozier.
Alongside Sir John were 128 officers, all of them aiming to find the elusive sea route linking the Atlantic to the Pacific via the Arctic seas off northern Canada.
The two vessels were last seen in the summer of that year bypassing whaling boats in Baffin Bay, off the coast of Greenland.
But soon afterwards, the ships vanished.
Inuit hunters told tales of starving white men who had been seen in the freezing wilderness over the following months and years.
Historians speculated that the ships had become trapped in the vast ice floes of the Canadian Arctic.
But until Thursday the exact fate of HMS Erebus had not been known.
The ship was discovered after the Canadian government ploughed millions of pounds into an extensive search operation.
It was found under just 11 metres of water near King William Island, 2,000km (1,200 miles) north-west of Toronto.
Canadian Prime Minister Stephen Harper even participated in the search himself, such was his determination to resolve the mystery.
HMS Terror has still not been discovered.
Canada has been attempting to assert sovereignty over the Northwest Passage in recent years, claiming the area as its own.
However the US government and others say it is international territory.

The 30-year-old Australian-born forward represented Scotland at the 2013 Rugby League World Cup.
He holds the record for the most consecutive appearances in the NRL with 215 between 2006 and 2014.
"The club has a great profile and stature in the game and I hope to be able to contribute to developing that wonderful culture," he said.
Douglas will join St Helens ahead of the 2017 season.
"I am told Saints have the best fans in the game and I can't wait to engage with them. I'm really looking forward to this next chapter in my life," he added.

Christina Haycock, 49, of Mold, Flintshire, caused council offices and courts to be evacuated, Caernarfon Crown Court was told.
She admitted making a bomb threat to Flintshire council on 22 January.
Defence counsel Jade Tufail said she wanted an apology "from those who were supposed to protect her at the time".
The court heard Haycock, a criminology and criminal justice graduate who had been in care from the age of three, told a council phone operator: "Shire Hall has got a bomb in it so I would tell them to get out."
Prosecutor Elen Owen said the call caused council staff, judges, magistrates and other court users in Mold to be evacuated with an estimated cost of tens of thousands of pounds.
Judge Philip Harris-Jenkins said: "In the circumstances of this case, because of the gravity of the offence, the disruption it caused, I am afraid I am not going to be in a position to suspend the sentence."

The crash was on the A96 Forres bypass.
It took place early on Tuesday evening.
The man's injuries are not thought to be life-threatening.
A shipwreck uncovered beneath the icy wastes of northern Canada has been identified as long-lost HMS Erebus.
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