The Saudi Press Agency, citing a statement by the Ministry, said that the drones caused the fire at the refinery in the city of Abqaiq in the Kingdom's oil-rich Eastern Province, as well as the blaze at the Khurais oil field, around 150 km from Riyadh.
Abqaiq is the location of the world's largest oil processing plant, which handles two-third of Saudi oil production. Khurais contains the country's second-largest oilfield.
The agency did not specify the type of drones involved or name the alleged perpetrator. The fires at both sites were brought under control and a probe was launched into the alleged attacks, reported the state media.
There was no immediate comment from Aramco or information about deaths or injuries. It was was also unclear how the attacks would impact oil production.
On the Houthi's Al-Masirah satellite news channel, military spokesman Yahia Sarie said the Iran-aligned rebels had launched 10 drones in a coordinated attack on the oil facilities in response to the Saudi-led war against rebel areas in Yemen.
He said Saturday's attack was "one of the biggest operations the Houthi forces had undertaken inside Saudi Arabia" and was carried out in "co-operation with the honourable people inside the Kingdom".
Multiple videos posted on social media earlier on Saturday showed an Aramco compound engulfed in flames and thick black smoke billowing from the site. In some vidoes, several loud bangs resembling the sound of explosions can be heard in the background.
Houthi fighters were blamed for drone attacks on the Shaybah natural gas liquefaction facility in August and on other oil facilities in May. The rebel movement is fighting the Yemeni government and a Saudi-led coalition.
Yemen has been at war since 2015, when President Abdrabbuh Mansour Hadi was forced to flee the capital Sanaa by the Houthis. Saudi Arabia backs Hadi and has led a coalition of regional countries against the rebels.
Saudi Arabia accuses Iran of arming and directing the Houthi rebels in Yemen to launch drone and rocket attacks across the kingdom's border.
( With inputs from IANS )