Saudi Arabia's Energy Minister Khalid al-Falih said on Thursday the kingdom would halt all oil shipments through the strait immediately.
The suspension will last "until the situation becomes clearer and maritime transit through Bab al-Mandeb is safe", al-Falih said in a statement.
Houthi rebels targeted Saudi tankers in the Red Sea, causing minor damage to one, the Saudi-Emirati coalition fighting the group in the war-torn country said.
A statement by the coalition said one tanker was attacked west of Yemen's Hodeidah port but did not name the vessel or describe how it was hit.
"The Saudi oil tanker was subjected to slight damage due to the attack by the Houthi militia," the statement said. "Thankfully the attack failed due to immediate intervention of the coalition's fleet."
A statement from Saudi Aramco said "two Very Large Crude Carriers [VLCCs], each with a two million barrels capacity ... were attacked by terrorist Houthi militia this morning in the Red Sea. One of the ships sustained minimal damage. No injuries nor oil spill have been reported".
Earlier on Wednesday, the Houthi-run Almasirah TV network reported rebels targeted a warship named the Dammam off the western coast of Yemen. The rebels later said they launched a missile attack on the vessel.
Bob Cavnar, an oil industry expert, told Al Jazeera the announcement had already added $1 to the cost of a barrel of oil on stock markets. He said any delay in the delivery of crude from Saudi Arabia would have repercussions on global supplies and economies.
"Even a small shortage - like two or three million barrels of oil a day - could have a substantial effect on crude and that causes an accordion effect throughout global economies. So it could have a long-term effect," said Cavnar.
"Clearly this is going to have an effect on global shipping. Any kind of danger to shipping and crews is going to cause a major disruption until the threat is neutralised."
The attacks and temporary suspension of oil transport could bring other countries into Yemen's devastating war, another analyst said.
"The Red Sea is a very important shipping lane. If there is a major disruption, European powers, Egypt and the United States would all have reason to intervene," energy consultant Ellen R Wald wrote in Forbes.
"They have significant interests in protecting the freedom of the seas through the passageway. An international intervention against the Houthis may be just what Saudi Arabia wants."
Battle for key port
On June 13, Yemeni government forces - backed by the military alliance - waged a wide-ranging operation to retake Hodeidah and its strategic seaport from the Houthis.
Government forces continued to advance towards the city and on June 19 they seized Hodeidah's international airport. But they have since been unable to make additional gains.
Saudi Arabia and the United Arab Emirate's main justifications for the intervention was to protect shipping routes such as the Red Sea, which is used to bring Middle Eastern oil and Asian goods to Europe through the Suez Canal. They said they foiled previous attacks in the Red Sea in April and May.
The coalition accuses the Houthis of smuggling weapons through Hodeidah's port.
Since the start of the Hodeidah offensive, more than 121,000 people have fled the city, according to a United Nations report.
Impoverished Yemen has been wracked by violence since 2014, when the Houthis overran much of the country, including the capital Sanaa.
The conflict escalated in 2015 when Saudi Arabia and its allies - who accuse the Houthis of serving as Iranian proxies - launched a massive air campaign in Yemen aimed at rolling back rebel gains.
The coalition has repeatedly accused regional rival Iran of arming the rebels, allegations the Houthis and Iran deny.
Yemen is also now the world's worst humanitarian crisis with more than 22.2 million people in need of assistance. Malnutrition, cholera, and other diseases have killed or sickened thousands of civilians over the years.