The royal court said in a statement that Abdullah, believed to be around 90, died at 1:00 am local time (0330 IST), expressing its "great sadness and mourning".
Salman, 79, had been defence minister and previously governor of the capital Riyadh.
Another of the late monarch's half-brothers, Moqren, was named the new crown prince.
Abdullah will be buried later today following afternoon prayers, and citizens would be invited to pledge allegiance to the new monarch and the crown prince at the royal palace, the statement said.
Under Abdullah, who took the throne in 2005, Saudi Arabia has been a key ally of Washington in the Arab world, most recently joining the US-led coalition carrying out air strikes against the Islamic State group in Syria and Iraq.
US President Barack Obama was quick to pay tribute to Abdullah as a valued ally.
"As our countries worked together to confront many challenges, I always valued King Abdullah's perspective and appreciated our genuine and warm friendship," Obama said in a written statement shortly after the monarch's death.
As the top producer in the Organisation of the Petroleum Exporting Countries, Saudi Arabia has been the driving force behind the cartel's refusal to slash output to support oil prices, which have fallen by more than 50 per cent since June.
The two main global crude contracts surged today following Abdullah's death, amid uncertainty over whether the new king would maintain that policy.
The kingdom is also home to Islam's holiest sites, Mecca and Medina, and its role as a spiritual leader for Sunni Muslims has seen it vying for influence with Shiite-dominated Iran.
Wary of the rising influence of Islamist movements, Saudi Arabia has been a generous supporter of Egyptian leader Abdel Fattah al-Sisi since the army's ouster of Mohamed Morsi of the Muslim Brotherhood.
It has also played a key role in supporting opposition to Iran-backed President Bashar al-Assad of Syria, allowing US troops to use its territory to train rebel fighters.
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