Prince Abdulaziz is a veteran of the energy ministry, which he joined in the 1980s.
Oil producers want to halt a slide in prices despite previous production cuts and U.S. sanctions on Iran and Venezuela.
European benchmark Brent was selling at US$61.54 per barrel Friday, in contrast with more than US$75 this time past year but up from around US$50 at the end of December 2018.
Brent was up 26 cents, or 0.4%, at $62.85 a barrel by 0349 GMT, while USA crude was 27 cents, or 0.5%, higher at $58.12 a barrel.
As energy minister, al-Falih faced challenging market conditions, including a boon in US shale production and global trade tensions that created an environment of lower oil prices.
Analysts say the OPEC+ group's Joint Ministerial Monitoring Committee, which monitors a supply cut deal reached past year, has limited options when it meets in Abu Dhabi on Thursday.
"His comments today suggest we shouldn't expect any major policy changes from the kingdom, which still wants to see oil inventories falling".
Al-Falih, who'd taken over the energy portfolio from longtime oil minister Ali al-Naimi, was tasked with playing a key role in Prince Mohammed's economic overhaul of the country. He was also in charge of the kingdom's industrial development.
Over the weekend, Alhajii tweeted that the new energy minister "comes to office with unprecedented experience, knowledge of oil market economics/politics".
Saudi Arabia's King Salman Bin Abdulaziz named his son Prince Abdulaziz as the country's energy minister, replacing Khalid Al-Falih, who led three years of active Opec (Organization of Petroleum Exporting Countries) diplomacy to forge a global alliance with producers such as Russian Federation to limit production in order to prop up prices.
His relationship with the crown prince will be critical.
The prince is King Salman's fourth son and an older half brother to the 34-year-old Crown Prince Mohammed bin Salman. He joined the energy ministry in 1987 as an adviser after a brief period working at King Fahd University of Petroleum and Minerals, where he received a degree in industrial engineering and an MBA.
He was promoted to deputy oil minister in 1995 and later became an assistant minister for petroleum affairs from 2004 to 2015. During that period he led a team responsible for Saudi Arabia's oil strategy as well as long term Opec strategy. "Energy efficiency programs will get a boost under his leadership".
Until Prince Abdulaziz's appointment late on Saturday night, Saudi Arabia's oil ministry has been since 1960 headed by civilian technocrats.
While Prince Mohammed's programme is ultimately created to wean Saudi Arabia's economy off its addiction to oil, in the short term he needs a higher oil price to finance his plans.
Still, that's considerably below the $80-$85 a barrel that analysts say is needed to balance Saudi Arabia's budget.
Aramco is preparing to sell up to a 5% stake by 2020-2021, in what could be the world's biggest IPO.
But the style of Saudi Arabia's oil-market management is likely to be different.
Trade and geopolitical tensions are affecting the market, Mazrouei said. Riyadh has targeted a valuation of US$2 trillion, but analysts and bankers have warned that this might be optimistic. The government has also been grappling with a widening budget deficit, which the International Monetary Fund projects will rise to 6.5 per cent of GDP this year.
Opec oil output rose in August, gaining for the first month this year as higher supply from Iraq and Nigeria outweighed restraint by Saudi Arabia and losses caused by USA sanctions on Iran, a Reuters survey found. Oil has remained stubbornly around US$60 a barrel.
Saudi officials have said they would not sign a deal that would deprive the kingdom of the possibility of enriching uranium or reprocessing spent fuel in the future - both potential paths to a bomb.
On Sunday, the United Arab Emirates' energy minister Suhail al-Mazrouei said OPEC and non-OPEC producers were "committed" to achieving oil market balance.
The prince will have a far narrower remit than his predecessor.