Qaboos enjoyed the reputation of being a moderate, able to cultivate good relations both with the sultanate's Sunni neighbours in the Gulf as well as with their rival, Shiite Iran.
Under Qaboos, Oman, a country of around 4.5 million people, remained unaffected by regional turbulence including violent militancy and a devastating Saudi-led war in neighbouring Yemen.
In his speech, Haitham, 65, said his country will keep calling for and contributing to solving differences through peaceful means.
"We will keep supporting the Arab League to achieve its goals ... and distance the region from conflicts and differences," he added.
Haitham, who is Qaboos' cousin and was serving as culture minister, was sworn in as Oman's new leader, hours after the death of 79-year-old Qaboos, who ruled the Gulf state for five decades.
Qaboos deposed his father in a bloodless coup in 1970 and was credited with transforming Oman into a modern country and an influential regional player.
A three-day national mourning was declared for Qaboos, whose death was confirmed by the royal court in an obituary carried by state media early Saturday.
A funeral was held and he was laid to rest in the cemetery of the ruling family in Muscat after funeral prayers were observed. The funeral was attended Sultan Haitham and Omani officials.
The cause of Qaboos' death was not specified, but there had been speculation concerning his health after he made a short trip to Belgium for medical reasons last month.
Qaboos left no children behind to take his place and had not publicly appointed a successor.
The Omani royal court announced "a state of mourning and the halt of official work in the public and private sectors for three days and the flying of flags at half-mast over the forthcoming 40 days."
Following his death, the country's Defence Council met, opened the will of the late sultan and read it out in the presence of members of the royal family.
"The letter declared that Haitham bin Tariq Al Said, Qaboos' cousin, is the new Omani sultan," state-run news agency ONA reported.
Tributes poured in from across the region and the world.
British Prime Minister Boris Johnson said he was "deeply saddned" by the death of Qaboos.
"He was an exceptionally wise and respected leader who will be missed enormously," Johnson said.
Dubai's ruler Sheikh Mohammed Bin Rashid Al Maktoum, who also serves as vice-president and premier of the United Arab Emirates, described Qaboos in a tweet as "the sultan of love and wisdom."
Kuwait's Emir Tamim bin Hamad Al Thani ordered official mourning for three days. All government institutions will shut down and Kuwait flag will fly at half-mast for three days, according to a government statement. The rulers of Qatar and Bahrain also declared three days of official mourning.
Jordan's King Abdullah offered his condolences, hailing Qaboos as a wise leader. "May God rest his soul in peace and grant the people of Oman patience and fortitude", he said in a tweet.
Saudi Arabia's royals offered their "heartfelt condolences" to the royal family and the Omani people.
"With the deepest sorrow King Salman and Crown Prince Mohammed bin Salman received the news of Sultan Qaboos," according to a royal court statement.
The Egyptian presidency praised Qaboos as a "wise leader who dedicated his life to his nation."
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