At the start of trading, Brent crude jumped 19% to $71.95 a barrel, while the other major benchmark, West Texas Intermediate, rose 15% to $63.34.
Prices eased back slightly after US President Donald Trump authorised the release of US reserves.
It could take weeks before the Saudi facilities are fully back on line.
The drone attacks on plants in the heartland of Saudi Arabia's oil industry included hitting the world's biggest petroleum-processing facility.
US Secretary of State Mike Pompeo said Tehran was behind the attacks. Iran accused the US of "deceit."