Footage released on Saturday showed a huge blaze at Buqayq, site of Aramco’s largest oil processing plant, while a second drone attack started fires in the Khurais oilfield.
The huge fires are now under control at both facilities, state media said.
A spokesman for the Ansarullah group in Yemen said it had deployed 10 drones in the attacks.
The military spokesman told al-Masirah TV that further attacks could be expected in the future.
“At 04:00 (01:00 GMT), the industrial security teams of Aramco started dealing with fires at two of its facilities in Buqayq and Khurais as a result of… drones,” the official Saudi Press Agency reported.
The Yemeni group has promised to widen the range of its attacks on Saudi Arabia, which has waged a war against Yemeni people since 2015.
“These attacks are our right, and we warn the Saudis that our targets will keep expanding,” spokesman Yahya Saree said in a statement read out on Al Masirah TV.
“We have the right to strike back in retaliation to the air strikes and the targeting of our civilians for the last five years.”
An official with Yemeni Ansarullah movement has warned that Houthi fighters will capture Riyadh, the Saudi capital, if KSA does not stop its invasion of Yemen
The Saudi-led war has killed tens of thousands of people and sparked what the United Nations calls the world’s worst humanitarian crisis.