Brent crude futures gained $1.44 to $65.04 a barrel by 0958 GMT, while US West Texas Intermediate crude was up $0.89 at $59 a barrel.
The attacks knocked down more than half of Saudi Arabia's crude production and severely limited the country's spare capacity, a cushion for oil markets in any unplanned outage.
"Global available spare capacity is extremely low at present following the weekend attacks, leaving little room for additional outages, which tends to be price supportive," UBS oil analyst Giovanni Staunovo said.
Earlier this week, Saudi Arabia set out a timeline for the resumption of full operations, saying it had restored supplies to customers at levels prior to the attacks by drawing from its oil inventories.
But it said it would restore its lost production by the end of this month and bring its output capacity back to 12 million barrels per day by the end of November.
"These plans suggest Saudi Arabia will have no spare capacity for at least the next two and a half months and therefore no way to absorb any further shocks," consultancy Energy Aspects said.
Saudi Arabia, the world's leading oil exporter, has said the crippling attack on its oil sites was "unquestionably sponsored" by regional rival Iran.
US President Donald Trump said there were many options short of war with Iran and added that he had ordered the US Treasury to "substantially increase sanctions" on Tehran. Iran has denied involvement in the strikes.
The head of the International Energy Agency said on Wednesday it saw no need to release emergency oil stocks as markets were well supplied.
Following the attacks, Kuwait's oil sector is on high alert and has raised its security to the highest level as a precaution, a Kuwaiti official said.
The US Federal Reserve cut interest rates again on Wednesday to help sustain a record-long economic expansion.
"The Fed rate cut is supporting risk sentiment, which tends to help crude as well," Staunovo said.
Separately, weekly data from the Energy Information Administration on US oil inventories provided a mixed snapshot. Stockpiles of crude in the United States, the world's largest oil producer, rose by 1.1 million barrels last week against analysts' expectations for a drop of 2.5 million barrels.
However, stocks in Cushing, Oklahoma, the delivery point for benchmark futures, fell to their lowest since October 2018.
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