Grant Shapps said the UK remained “undaunted” after the assault by the Iran-backed rebel group, adding that the government’s commitment to protecting innocent lives and the freedom of navigation was “absolutely unwavering”.
None of the crew were injured and the HMS Diamond was undamaged, the Ministry of Defence said in a statement.
The ship shot the drone down using its missile defence system.
It is not the first time the HMS Diamond has been targeted by the group. An attack earlier this month is widely seen as the step which triggered joint US-UK airstrikes on Houthi positions in Yemen, following months of provocation.
The militants claim they are targeting ships linked to Israel as part of a campaign of support for Palestinians during the war in Gaza.
This assertion has been rejected by the UK and the US, however.
In the early hours of Saturday, the US also destroyed a Houthi anti-ship missile, just hours after an attack that left a fire raging on a British-linked oil tanker.
Mr Shapps condemned that assault as “intolerable”, while the government said Britain and its allies reserved “the right to respond appropriately”.
Despite the strikes by the allies, the defiant Houthis have vowed bloody retaliation and have continued to attack vessels in the Red Sea and beyond.
The MoD said: “These intolerable and illegal attacks are completely unacceptable and it is our duty to protect the freedom of navigation in the Red Sea.
“We want to thank the brave crew of HMS Diamond for their service to defend British and international interests.”