It came after the Huthis vowed to keep up their attacks despite repeated US and British strikes against them.
One missile missed its target and a US Navy destroyer shot down the other two, National Security Council spokesman John Kirby said.
The continuing Huthi action "means we're obviously still going to have to do what we have to do to protect that shipping," he added.
US Central Command said the missiles were fired "toward the US-flagged, owned, and operated container ship M/V Maersk Detroit" without mentioning a second vessel being targeted.
No injuries or damage to the ship were reported, CENTCOM added.
Danish shipping giant Maersk earlier said two ships belonging to a US subsidiary and bound for the Red Sea turned back after hearing explosions while transiting the Bab al-Mandeb strait between the Horn of Africa and the Arabian Peninsula.
A US navy escort accompanying the Maersk Detroit and the Maersk Chesapeake also "intercepted multiple projectiles", the company said.
"The crew, ship, and cargo are safe and unharmed. The US Navy has turned both ships around and is escorting them back to the Gulf of Aden," it added.
The Huthis later confirmed a "clash" had taken place with American warships while they were protecting the two US commercial ships.
Their military spokesman Yahya Saree claimed several of the group's missiles hit their targets, forcing the US merchant ships to turn back.
- New shipping disruptions -
United Kingdom Maritime Trade Operations, a security agency run by Britain's navy, said it had received reports of "an explosion approximately 100 metres" from a vessel 50 nautical miles south of the Yemeni port of Mokha, which overlooks Bab al-Mandeb.
British maritime risk management company Ambrey corroborated the UKMTO and Maersk reports, adding that both vessels had last called in Oman.
Huthi attacks since mid-November have disrupted trade in the Red Sea, which connects Europe and Asia and carries around 12 percent of international maritime traffic.
The rebels say they are targeting Israeli-linked vessels in support of Palestinians in Gaza, which has been ravaged by the Hamas-Israel war that has inflamed tensions across the Middle East.
Saree reiterated Wednesday that the Huthis would continue targeting Israeli-linked shipping "until a ceasefire is achieved in Gaza".
Several shipping firms have diverted away from the Red Sea, instead taking the longer and more expensive route around the Cape of Good Hope in South Africa.
It follows difficult years for the industry during the Covid-19 pandemic, when freight rates reached unprecedented levels due to blockages in supply chains.
The United States and Britain have carried out two rounds of joint strikes this month aimed at reducing the Huthis' ability to target shipping.
The US military has also launched a series of unilateral air raids on the rebels' missiles.
It said its latest strikes early on Wednesday destroyed two Huthi missiles that posed an "imminent threat" to ships in the area.
The Huthis have reacted to the US and UK strikes with defiance, firing at more ships and declaring American and British interests to be legitimate targets.
Washington is also seeking to put diplomatic and financial pressure on the Huthis, redesignating them as a terrorist organisation last week after dropping that label soon after President Joe Biden took office.
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