A tanker with links to the UK was on fire for several hours in the Gulf of Aden after a being hit by a missile fired by the Houthis.
The Iran-backed movement, based in Yemen, said it targeted the Marlin Luanda on Friday in response to "American-British aggression".
The US and UK have launched air strikes on Houthi targets in response to attacks on ships in the Red Sea region.
French, Indian and US naval ships provided assistance to the vessel.
The operator of the Marlin Luanda is registered as being Oceonix Services Ltd, a UK registered company.
The tanker flies under the flag of the Marshall Islands and is operated on behalf of Trafigura - a multinational trading company.
In an update on Saturday, Trafigura said all crew were safe and that the fire in a cargo tank had been put out. The ship was now sailing towards a safe harbour, the company said.
It is the latest attack on commercial shipping by the Houthis in and around the Red Sea. The group says it is targeting vessels in the region in support of Palestinians in Gaza, where Israel is fighting Hamas.
In a statement, a Houthi spokesperson claimed the Marlin Luanda was a British ship and was targeted in response to "American-British aggression against our country".
The UK government said attacks on commercial shipping are "completely unacceptable" and that Britain and its allies "reserve the right to respond appropriately".
The UK Maritime Trade Operations (UKMTO) said the incident happened 60 nautical miles south-east of Aden.
It warned other vessels to transit with caution and report any suspicious activity.
Later, the US Central Command said its forces had conducted a strike at 03:45 local time (00:45 GMT) on Saturday "against a Houthi anti-ship missile aimed into the Red Sea and which was prepared to launch". They "destroyed the missile in self-defence", CentCom said.
Since November, the Houthis have launched dozens of attacks on commercial vessels travelling through the Red Sea, one of the world's busiest shipping lanes.
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