Arriving at the summit just before 1am BST, the veteran explorer became the first man to cross the north and south poles and climb the world's highest peak.
At 65, he also becomes the oldest Briton and the first British pensioner to climb Everest.
During his acclimatisation, he said: "Forget about thinking you are going to succeed. You have just got to keep plodding. Imagine it's a mountain with no top."
The record-breaking adventurer, who hopes to raise £3 million for cancer charity
Marie Curie
, had a triple heart bypass in 2003 following a heart attack.
During his 2005 attempt on Everest, Sir Ranulph had a heart attack at 8,500 metres and last year was forced to turn back at 8,400 metres after suffering from exhaustion.
After his second attempt he said: "I won't be returning to Everest. It's a seven week trip - last time I had a heart attack, this time bad timing and weather scuppered my chances. I think any third attempt would be bad luck."
However, his change of mind meant he finally conquered the 8,850 metre peak.
Among Sir Ranulph's numerous achievements are becoming the first man to reach the north and south poles by land unaided, completing seven marathons on seven continents in seven days in 2003 and climbing the treacherous north face of the Eiger in 2007.
Sir Ranulph also joined solo yachtsman Sir Robin Knox-Johnston and John Simpson, the BBC's world affairs editor, for the BBC Two series Top Dogs which saw the trio file war reports from Afghanistan, trek across the Arctic and sail around Cape Horn.
Speaking to the Press Association, Simpson said he was thrilled and extremely proud of his "close friend".
"I want to express my sheer delight and pride in his fantastic achievement," he said. "I knew he would do it on the third attempt. We have already broken out the Champagne.
"My son is named after him and this morning we will put a tiny drop in his milk so he can celebrate too."
Sir Ranulph said: "I have summited Everest
for Marie Curie Cancer Care
which has long been a personal goal.
"Thanks to my sponsor Brewin Dolphin for making this possible. I urge everyone who followed my attempt last year to give generously to Marie Curie so that we can at last achieve our £3 million target to support its pioneering work in end-of-life care."
Sir Ranulph apparently began his latest assault on Everest three weeks ago, but chose to keep the expedition low-key after two unsuccessful attempts in 2005 and 2008.
A spokeswoman for Marie Curie said: "He wanted to do it quietly with no fuss. He only just failed last year - he got to within a short distance of the summit but had to turn back from exhaustion.
"So he decided he wanted to keep it very low-key without all the build-up and media attention of last year."
Sir Ranulph, who has already raised millions for Marie Curie Cancer Care, lost his first wife, sister and mother to cancer within 18 months.
The charity's chief executive Thomas Hughes-Hallett said: "Everyone at Marie Curie is delighted that Sir Ranulph has conquered Everest. We know it meant so much to him - we are so grateful to Ran for all his support for the charity and for his determination to personally take the Marie Curie flag to the summit."
Sir Ranulph's triumphs and tribulations
By Tom Rayner, Press Association
Described by the Guinness Book of Records as the world's greatest living explorer, Sir Ranulph Fiennes has refused to let a double heart bypass dent his spirit of adventure.
Everest held painful memories for the quintessential English hero. In 2005 the adventurer suffered a heart attack during an unsuccessful attempt to scale the mountain.
But the same indomitable spirit that allowed Sir Ranulph to run seven marathons in seven consecutive days in 2003 saw him return to Everest in 2008 only to fail in his bid to reach the summit, before finally succeeding early today at the third attempt.
But 65-year-old Sir Ranulph's motives are not purely driven by personal ambition and his expeditions have already raised millions for charity Marie Curie Cancer Care.
As well as his own medical set backs Sir Ranulph had to overcome the death of his wife Virginia, known as Ginny, after a short battle with cancer in February 2004 aged just 56.
The pair were childhood sweethearts and met when he was just 12 years old.
They married in 1970 and Sir Ranulph once said that his marriage was the greatest achievement in his remarkable life.
Sir Ranulph's doughty character was formed by an early childhood spent in South Africa before being sent to top public school Eton aged 12.
A promising Army career with the Royal Scots Greys, and later the SAS, was cut short in 1968 when he was dismissed for blowing up the set of the film Dr Doolittle in a prank that went wrong.
However, it was after leaving the army that Sir Ranulph achieved his full potential and circumnavigated the globe from Pole to Pole in a three-year, 35,000 mile trek. He was the first man to reach both Poles by surface means.
Sir Ranulph was also part of a team that discovered the lost city of Ubar, described by Lawrence of Arabia as the Atlantis of the Sands.
In 1993 Sir Ranulph was an awarded an OBE by the Queen for human endeavour. An endeavour never better demonstrated than during his attempt at a solo expedition to reach the North Pole unsupported.
Sir Ranulph plunged his hand into icy water to retrieve a lost sledge and suffered extreme frost bite. As a result of the injury he lost all of the fingers on his left hand below the knuckle.
Over the years Sir Ranulph has raised millions of pounds for medical research - particularly for multiple sclerosis and breast cancer.
Sir Ranulph's cousins, Hollywood actors Joseph (Shakespeare in Love) and Ralph Fiennes (The English Patient), will be celebrating their relative's latest accomplishment.
Prince Charles once said that his admiration for Sir Ranulph was "unbounded" and it remains to be seen what more this man can achieve in his life.
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