Since he took the throne in 2005, Riyadh has been a key Arab ally of Washington, last year joining the coalition carrying out air strikes against the Islamic State jihadist group.
World leaders praised the king as a key mediator between Muslims and the West, but campaigners criticised his rights record and urged Salman to do more to protect freedom of speech and women's rights.
Gulf rulers and leaders including Turkish President Recep Tayyip Erdogan and Pakistani Prime Minister Nawaz Sharif were among those who attended Abdullah's simple funeral at Riyadh's Imam Turki bin Abdullah mosque on Friday.
The late king's body, wrapped in a cream-coloured shroud, was borne on a simple litter by members of the royal family wearing traditional red-and-white checked headgear.
The body was quickly moved to nearby Al-Od public cemetery and buried in an unmarked grave, in keeping with tradition.
Palestinian President Mahmud Abbas and Malaysia's Prime Minister Najib Razak arrived later to deliver condolences.
In the evening hundreds of Saudis queued to enter a royal palace where they rubbed cheeks and kissed the hands of their new leaders, in a symbolic pledge of allegiance.
Officials did not disclose the cause of Abdullah's death, but the long-ailing ruler had been hospitalised in December with pneumonia.
- Valued ally -
President Barack Obama was quick to pay tribute to Abdullah on Friday as a "valued" ally, and the State Department said Washington saw no indication that cooperation would change.
Biden said on Twitter he would lead a delegation to Saudi Arabia "to pay respect and offer condolences".
Salman vowed Friday to keep the conservative, oil-rich Muslim kingdom on a steady course and moved to cement his hold on power.
In his first public statement as king, Salman, 79, vowed to "remain, with God's strength, attached to the straight path that this state has walked since its establishment".
He called for "unity and solidarity" among Muslims and vowed to work in "the defence of the causes of our nation".
Moving to clear uncertainty over the transition to the next generation, he named his nephew, Interior Minister Prince Mohammed bin Nayef, 55, as second in line to the throne behind Crown Prince Moqren, 69.
That helps to solidify control by his Sudayri branch of the royal family.
Salman also appointed one of his own sons, Prince Mohammed, as defence minister of the world's top oil exporter and the spiritual home of Islam.
As the top producer in the Organization 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 percent since June.
Prices surged Friday, due to uncertainty over whether the new king would maintain that policy, but the International Energy Agency's chief economist said he did not foresee major policy shifts.
"And I expect and hope that they will continue to be a stabilisation factor in the oil markets," Fatih Birol told AFP in Davos, Switzerland.
- Royal family stalwart -
Saudi Arabia is 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.
Tehran nonetheless offered its condolences, saying Foreign Minister Mohammad Javad Zarif would be among those travelling to Riyadh to take part in official ceremonies Saturday.
Behind his thick, jet-black moustache and goatee, Abdullah had a shrewd grasp of regional politics.
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 ousted Mohamed Morsi of the Muslim Brotherhood.
Egypt declared seven days of official mourning for Abdullah.
Riyadh has also played a key role in supporting opposition to Iran-backed President Bashar al-Assad of Syria, and will allow US troops to use its territory to train rebel fighters.
Salman is widely expected to follow closely in Abdullah's footsteps, in foreign and energy policy as well as in making moderate reforms.
Abdullah pushed through cautious changes, challenging conservatives with such moves as including women in the Shura Council, an advisory body.
He promoted the kingdom's economic development and oversaw its accession to the World Trade Organization, tapping into the country's massive oil wealth to build new cities, universities and high-speed railways.
But the country is still strongly criticised for a dismal human rights record, including the imprisonment of dissidents. It is also the only country in the world that does not allow women to drive.
Amnesty International head Salil Shetty told AFP that "the Saudi regime seems insensitive to human rights and human dignity".
And while Saudi Arabia has managed to avoid the social upheaval that has shaken many of its neighbours in recent years, thanks in large part to massive public spending, the new king will face some major challenges, especially as falling oil prices cut into state revenues.
Since the death in 1953 of the kingdom's founder, King Abdul Aziz bin Saud, the throne has passed systematically from one of his sons to another.
Abdul Aziz had 45 recorded sons. Abdullah, Salman and Moqren were all born to different mothers.