The projectiles were headed towards international shipping lanes in the southern Red Sea, the United States said.
Britain's defense minister said it was the largest attack in the area by the militants to date as the three-month-long war between Israel and Hamas in Gaza spills over into other parts of the Middle East.
U.S. Central Command said 18 drones, two anti-ship cruise missiles and one anti-ship ballistic missile were shot down by the two navies in the incident.
No injuries nor damage were reported, it said, adding that this was the 26th Houthi attack on commercial shipping lanes in the Red Sea since November 19.
Houthi military spokesperson Yahya Sarea, gave a televized speech on Wednesday, saying the attack had targeted a U.S. ship it said was providing support to Israel.
The operation was a response to a previous U.S. attack that killed 10 Houthi militants, he added.
The Houthis control most of Yemen and have been targeting the route to show their support for the Palestinian Islamist group, Hamas.
The attacks have severely disrupted international commerce on the key route between Europe and Asia, which accounts for about 15% of the world's shipping traffic.
Many shipping companies including German shipping group Hapag Lloyd and its Danish rival Maersk have been forced to reroute their vessels, taking the longer journey around Africa.
The Iranian-backed Houthis have vowed to continue attacks until Israel halts the conflict in Gaza, and warned they would would attack U.S. warships if the militia group itself was targeted.
