Saudi Arabia is vulnerable to possible Iranian reprisals after Tehran vowed "revenge" following the strike on Friday that killed powerful commander Qasem Soleimani in Baghdad.
"In light of the rapid developments, the kingdom stresses the importance of exercising restraint to guard against all acts that may lead to escalation, with severe consequences," the official added.
Saudi Arabia's foreign ministry echoed a similar call for restraint at the weekend and King Salman emphasised the need for measures to calm tensions in a phone call on Saturday with Iraqi President Barham Saleh.
In a separate phone call with Iraq's Prime Minister Adel Abdel Mahdi, Saudi Crown Prince Mohammed bin Salman stressed "the need to make efforts to calm the situation and de-escalate tensions", the official Saudi Press Agency reported.
The crown prince has instructed Prince Khalid bin Salman, his younger brother and deputy defence minister, to travel to Washington and London in the next few days to urge restraint, the pan-Arab Asharq al-Awsat newspaper reported.
Prince Khalid will meet White House and US defence officials, the paper said, citing unnamed sources.
The killing of Soleimani, seen as the second most powerful man in Iran, is the most dramatic escalation yet in spiralling tensions between Washington and Tehran and has prompted fears of a major conflagration in the Middle East.
US President Donald Trump, who ordered the drone strike, has warned that Washington will hit Iran "very fast and very hard" if the Islamic republic attacks American personnel or assets.
Saudi Arabia and the United Arab Emirates, both allies of Washington, are also vulnerable to Iranian counter strikes, analysts say.
A string of attacks attributed to Iran has caused anxiety in recent months as Riyadh and Washington deliberated over how to react.
In particular, devastating strikes against Saudi oil installations last September led Riyadh and Abu Dhabi to adopt a more conciliatory approach aimed at avoiding confrontation with Tehran.
Analysts warn that pro-Iran groups have the capacity to carry out attacks on US bases in Gulf states as well as against shipping in the Strait of Hormuz -- the strategic waterway that Tehran could close at will.