The company has not given a timeline for the resumption of full output.
The drone attacks on plants in the heartland of Saudi Arabia's oil industry included hitting the world's biggest petroleum-processing facility. "If anything, the risk of tit-for-tat regional escalation that pushes oil prices even higher has gone up significantly".
Iran has denied USA allegations it launched the assault as tensions remained high over Tehran's collapsing nuclear deal.
The remarks came as the price of oil on the world market soared in response to the attacks.
The U.S has previously accused Iran's hand in the attacks on two oil tankers in the Gulf in June although Tehran rejected them as "wild allegations". However, Saudi Arabia has not yet blamed any party for Saturday's strike. The reserve holds about 630 million barrels.
Secretary of State Mike Pompeo blamed Iran at the weekend, prompting Tehran to accuse the USA of deceit.
During the weekend, the International Energy Agency stressed that global markets would not be disrupted by the Saudi attacks.
According to the USA official, 17 structures at Abqaiq suffered damage on their west-northwest facing sides, along with two points of impact at Saudi's Khurais facility.
"However Saudi Aramco keeps some redundancy in the system to maintain production during maintenance", Rapidan added, meaning operations could return to pre-attack levels sooner.
The strikes knocked out more than 5% of global oil supply.
The drone attacks led to the interruption of an estimated 5.7 million barrels of crude oil.
The attack came after Trump said a meeting with Iranian President Hassan Rouhani was possible at the UN General Assembly in NY this month. Tehran ruled out talks until sanctions are lifted.
FILE - In this photo released by the official website of the office of the Iranian Presidency, President Hassan Rouhani speaks at a cabinet meeting in Tehran, Iran, Sept. 11, 2019. The Saudi War in Yemen is going extremely poorly, and the ability of the Houthis of all people to launch drone attacks into key Saudi sites without air defense stopping them suggests this war could quickly get very costly.
Meanwhile, an Iranian Islamic Revolutionary Guard Corps commander said that all USA military bases and vessels in the region are within the reach of Iranian missiles, Iran's Tasnim News Agency reported on Sunday.