His death was announced by police after Cook, 59, was airlifted by helicopter from Ben Stack in the Highlands following his collapse on the rugged peak during his summer holiday.
Police said Cook, a keen walker, was nearing the top of Ben Stack, a conical mountain that stands 721 metres high, with his second wife Gaynor when he became ill at 2.23pm.
An air-sea rescue helicopter was dispatched to take him to hospital in Inverness, where he was declared dead at 4.05pm, five minutes after his arrival in the emergency ward, a spokesman for Raigmore hospital said.
Britain's domestic Press Association said it seemed he had suffered a heart attack.
In a statement, Blair paid tribute to Cook as "an outstanding, extraordinary talent - brilliant, incisive in debate, of incredible skill and persuasive power".
Blair lauded Cook's contribution
to British politics and Labour
"Though we disagreed over Iraq, I always respected the way in which he put his case... His contribution to the politics of Britain was hugely significant. His contribution to the Labour Party was immense. He will be missed," Blair said.
During his four years at the Foreign Office, Cook forged an "ethical" foreign policy for Britain, and supported Nato's 1999 intervention in war-torn Kosovo to wrest the mainly ethnic Albanian province from Serbia's control.
He quit Blair's government, as leader of the House of Commons, two days before the US and British invasion of Iraq in March 2003, saying: "I cannot support a war without international agreement or domestic support."
He easily won re-election in his central Scotland constituency of Livingstone in the general election last May that put Blair and Labour back in power for a third straight term.
Cook resigned in protest against
the invasion of Iraq
No longer a cabinet minister, he was a prolific commentator in the press, and many political analysts expected him to make a comeback after the anticipated resignation of Blair to make way for Chancellor of the Exchequer Gordon Brown.
Born near Glasgow, the only child of a science teacher, Cook first won election to parliament in 1974, and soon made a name for himself with his political flair.
He held a number of senior posts when Labour was in opposition to the Conservative Party, including shadow health and social security secretary, shadow foreign secretary and chairman of the Labour Party.
When Labour swept to victory in 1997, Blair promptly appointed Cook as his foreign secretary, and in that job he played a key role in Britain's military interventions in Kosovo alongside Nato and in war-racked Sierra Leone.
"His contribution to the Labour Party was immense. He will be missed"
British Prime Minister Tony Blair
His roughest moment as foreign secretary involved his personal life, when the tabloid press revealed that Cook had a mistress, Gaynor Regan, who was living in his London flat.
Blair told Cook to end either the relationship or his 28-year marriage. He opted for the latter, dumping his first wife Margaret, the mother of his two sons, at the airport in London as they were about to depart on holiday.
In 2001, Cook, who seemed to have lost interest in the position of foreign secretary, became leader of the House of Commons - an important but lower profile cabinet post that involves organising the government's legislative agenda.
He threatened to resign from the government early in 2003 over Blair's decision to keep the House of the Lords, the upper house of parliament, an all-appointed chamber.
But it was the impending US-led invasion of Iraq - without a clear United Nations mandate - which ultimately caused Cook to resign, in one of the most powerful speeches to be heard in the Commons in years.
"Why is it now so urgent that we should take military action to disarm a military capacity that has been there for 20 years, and which we helped to create?" he asked.
When his memoirs were published, Cook dropped a bombshell by suggesting that Blair knew all along that Iraq under Saddam Hussein had no weapons of mass destruction, the ostensible reason for launching the war.