Mr Pompeo had rejected claims by Yemen’s Iran-backed Houthi rebels that they had carried out the attacks.
Saudi Arabia said Saturday’s strikes had suspended the production of 5.7 million barrels of oil a day.
All eyes will be on the markets when they reopen on Monday, with experts expecting a significant rise in price.
Since ousting Yemen’s president in 2015, the Houthis have been fighting a Saudi-led and Western-backed coalition.
They said they had deployed 10 drones to attack the Abqaiq processing plant and the Khurais oilfield early on Saturday.
But Mr Pompeo said there was “no evidence” the drones came from Yemen and instead accused Iran.
The US has blamed Iran for other attacks on oil supplies in the region this year, amid continuing tension following President Donald Trump’s decision to reinstate sanctions after abandoning the landmark international deal which limited Tehran’s nuclear activities.
What has Iran said about the US accusations?
Mr Zarif took to Twitter to deride the US secretary of state, saying that “having failed at max pressure, Sec Pompeo’s…