Flames and smoke could be seen billowing from the key sites at Buqyaq and Khurais after the assaults early on Saturday morning.
Houthi rebels, who have been waging a five-year war with Saudi led forces in their Yemeni homeland, claimed responsibility for the “massive offensive operation” and said 10 drones were used in the “bombing”.
The group’s military spokesman Yahia Sarie, in a televised address on Al Masirah TV, warned of further action if the conflict in Yemen continues.
“The only option for the Saudi government is to stop attacking us,” he said.
However, Mr Pompeo has said there is “no evidence the attacks came from Yemen”.
Taking to Twitter, he accused Iran of an “unprecedented attack on the world’s energy supply”.
He added: "Tehran is behind nearly 100 attacks on Saudi Arabia while [President] Rouhani and [Foreing Affairs Minister] Zarif pretend to engage in diplomacy. Amid all the calls for de-escalation, Iran has now launched an unprecedented attack on the world's energy supply. There is no evidence the attacks came from Yemen. We call on all nations to publicly and unequivocally condemn Iran’s attacks.
“The United States will work with our partners and allies to ensure that energy markets remain well supplied and Iran is held accountable for its aggression”.
Saudi Arabia, which leads a coalition that intervened in Yemen in 2015 against the Houthis, has blamed regional rival Iran for previous attacks – allegations Tehran denies.
The kingdom’s interior ministry said in a statement that the fires were under control and an investigation into the attack was underway.
There are not believed to have been any injuries in the attack.
Saudi Aramco, the state-owned oil giant, described its Abqaiq oil processing facility in Buqyaq, around 200 miles northeast of the capital Riyadh, as “the largest crude oil stabilisation plant in the world”. Estimates suggest it can process up to seven million barrels of crude oil a day.
It has previously been targeted by militants, including a failed suicide bomber mission by al-Qaeda in February 2006.
Following the attack the White House said the United States was committed to keeping oil markets well-supplied in the wake of the attack, which sources said affected 5 million barrels per day of crude production - close to half of the kingdom's output or 5 per cent of global supply.
Additional reporting by Associated Press
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