The crown prince, aged 80 according to government records, served as the oil kingpin's defence minister for nearly five decades and had been in the United States since mid-June for medical treatment. He was operated on in July.
A half brother of King Abdullah, Sultan spent long periods abroad for undisclosed medical treatment.
A Western diplomat, speaking to AFP on condition of anonymity, said Prince Sultan had been hooked up to life support systems at New York Presbyterian hospital and was declared "clinically dead" more than a month ago.
Condolences flooded into the kingdom from world leaders as news of the prince's death was confirmed.
US President Barack Obama expressed "great regret" over the death of the prince, whom he described as a "strong supporter of the deep and enduring partnership between our two countries forged almost seven decades ago."
"On behalf of the American people, I extend my deepest condolences to King Abdullah, the royal family, and the people of Saudi Arabia."
In Jordan, King Abdullah II opened a World Economic Forum with a minute of silence in the late prince's honour, hailing him as a "champion of the Arab and Muslim cause."
Britain's Prime Minister David Cameron said he was "saddened" to hear of Sultan's death.
"He had many friends in this country, and we have all benefited from his wisdom and expertise in international affairs over his long years of service," the prime minister said.
Syrian President Bashar al-Assad, despite strains in ties with Riyadh over his security forces' deadly crackdown on anti-regime protests, also sent condolences to the king.
And Iranian Foreign Minister Ali Akbar Salehi sent condolences, expressing "sympathy to his counterpart and the deceased's next of kin," media reported.
Tensions between Sunni Saudi Arabia and Shiite Iran are running high as a result of US allegations that Iranian officials plotted to assassinate the Saudi ambassador to Washington.
Though Iran has fiercely denied the accusations, Saudi Arabia has accepted them and warned it was mulling "a suitable response."
Sultan died while King Abdullah, himself 87, was in hospital in Riyadh, recovering from an operation on his back on Monday.
Later on Saturday, the royal palace announced that the king had left the hospital and returned to the palace, where he would undergo further treatment.
The king underwent an operation in November for a debilitating herniated disc complicated by a blood clot, that put pressure on his spine, and he underwent further surgery in December.
The latest operation was aimed at repairing a slackening ligament near the third vertebra that state media said was successful.
The advanced age and failing health of the king and of his half-brothers in line to succeed him have raised concerns about the future of the oil giant in the face of the turmoil rocking the Arab world.
The Al-Saud family has ruled Saudi Arabia ever since the kingdom's establishment in 1932 and, under its rules of succession, the throne passes in turn to the sons of its founder Abdul Aziz, all of whom are now elderly.
Prince Nayef, who is expected to become crown prince, is 78 and was named second deputy premier in March 2009, filling a potential void in the succession lineup.
Another half-brother of King Abdullah, he holds the interior portfolio.
"There is only one candidate who has all the possibilities of becoming crown prince: Prince Nayef," Abdul Aziz al-Sager, chairman of the Gulf Research Centre, told AFP.
Khaled Dakhil, a political science professor at King Saud University, said the "most prominent candidate to inherit the throne is Prince Nayef."
"The choice should not take long, if it hasn't been made already, because the members of prince Sultan's family knew his health was very bad."
Except for a few demonstrations by the Shiite minority in eastern Saudi Arabia, the Sunni-dominated absolute monarchy has been largely spared the wave of pro-democracy protests sweeping other Arab countries.
But it has felt increasingly threatened by the growing influence in the region of its Shiite arch-rival Iran, which Riyadh accuses of sowing sectarian tensions.
In March, Riyadh spearheaded a military intervention by Sunni-ruled Gulf states in Bahrain as its Sunni ruling family crushed month-long pro-democracy protests led by the island state's Shiite majority.
Prince Sultan's funeral is to be held on Tuesday after his body has been repatriated from the United States.
Sultan had battled colon cancer since 2004, going to Switzerland and then the United States to seek treatment, according to diplomats.
His long illness and absences abroad held up important government decisions while raising questions about how the monarchy would take shape in the next generation of the Al-Saud family.