Oil inventories in the industrialized world—a proxy for the glut—have fallen to their lowest levels in two years. Brent crude, the international benchmark, rose past $65 a barrel in volatile trading Tuesday for the first time since June 2015 before falling to $63.46 a barrel on Tuesday evening in London.

Oil-industry players are now preparing for a more stable price around $60 a barrel or higher, after a three-year roller coaster when the price fell from over $100 a barrel in 2014 to less than $28 a barrel in 2016 before stagnating in the $40s and $50s much of this year.

“It’s actually tempting us to revise our price forecasts higher,” said Amrita Sen of consultancy Energy Aspects, which predicts Brent will average $64 a barrel next year.

The rise in oil prices comes with a quirk that could help U.S. companies: American oil prices haven’t gained as fast as Brent and remain more than $6 a barrel cheaper, at about $57 a barrel. That makes American oil more attractive to ship globally. The last time the price differential was this wide, in October, U.S. exports climbed to 2 million barrels a day, a record, according to German bank Commerzbank.

“This is definitely good news for U.S. exporters, who have already been ramping up” sales abroad, said Tom Pugh, a commodities economist at Capital Economics.

U.S. oil exporters and shale drillers have been locking in sales of their crude at higher prices just as American production is forecast to reach record levels in 2018.

The oil-price surge gives big Western oil companies like BP PLC, Exxon Mobil Corp.XOM +0.56% and Royal Dutch Shell PLC more financial breathing room than they have had in years. Many have already taken steps that show they can make money at prices below $60 a barrel. France’s Total SA on Tuesday scrapped the discount it will offer shareholders that opt to receive their dividend payment in stock—a signal that the company is more confident it can haul in enough cash to pay investors.

Meanwhile, Saudi officials on Tuesday said Saudi Arabian Oil Co. would invest $414 billion over the next decade, investments they said were needed for the kingdom to maintain its world-leading ability to produce 12 million barrels a day.

The most recent jolt to the market came from Great Britain, where one of Europe’s most important pipeline systems sprung a leak last week. About 450,000 barrels a day of North Sea oil production is shut off for the next two to three weeks for repairs to the Forties Pipeline System, said its owner, Ineos, the British chemicals and refining company.

“When you have big voluntary cuts, you can’t afford many unplanned outages,” said Olivier Jakob, managing director of Petromatrix, an oil research firm in Zug, Switzerland.

The oil-supply cuts and disruptions have coincided with healthy oil demand this year as the global economy enjoys a rare spurt of synchronized growth.

To be sure, oil prices are likely to correct downward after the U.K. pipeline system returns sometimes this month. Higher prices could cause the OPEC production deal to collapse as its members try to cash in on higher prices by releasing more output.

Analysts have warned that higher prices could cause a flood of new production from the U.S. In the third quarter, U.S. oil production for 2018 was hedged at its highest levels since 2014, allowing producers to keep increasing output even if prices fall next year, Citigroup said.

The volatility was on display Tuesday morning when prices suddenly nose-dived and fell below $65. For some the level represents a psychological ceiling at which they want to cash in and sell.

“The question is where does the next bull buyer come from,” said Michael Tran, director of energy strategy at RBC Capital Markets.

But the sheer amount of oil coming off the market in recent weeks may be too much even for the U.S. to make up for. Shale producers are under pressure to show profits instead of just pumping as much as possible.

Oil companies are working to fix the disruptions, but it isn’t clear how quickly.

Ineos had evacuated local residents and sent a team of engineers to the site of pipeline damage to assess how to fix a crack that had spread over the past several days. “It is expected to be a matter of weeks rather than days,” a company spokesman said.

The 590,000-barrel-a-day Keystone oil pipeline, an important oil artery between Canada and the U.S., has been transporting oil at 20%-reduced rates as a precautionary measure after pipe damage in South Dakota that forced its shutdown late November.

In Iraq, some 275,000 barrels a day of output remains shut down in Kirkuk oil fields, said Antoine Rostand of Paris-based consultancy Kayrros, which uses satellite data to assess output levels at individual facilities.

In Venezuela, a debt dispute with China and U.S. sanctions has knocked its export volumes down by 200,000 barrels a day in late October and November, according to Kpler, a tanker-tracking firm.

—Neanda Salvaterra, Sarah Kent and Summer Said contributed to this article.

Write to Georgi Kantchev at georgi.kantchev@wsj.com and Benoit Faucon at benoit.faucon@wsj.com