In a separate statement, Saree said the Houthis carried out missile and drone strikes on the Yemeni Red Sea port city of Mocha.
The port, controlled by the Aden-based government, is close to the strategic Bab el-Mandeb strait, which links the southern Red Sea with the Gulf of Aden and the Indian Ocean.
Should it be closed, Saudi Arabia would be deprived of an alternative to the Strait of Hormuz.
Three Yemeni government sources told Reuters that air defences were engaging drones launched by the Houthis towards the commercial quay and storage in the port.
Saree said the Houthi strikes targeted Saudi troops and weapons arsenal in the area. There was no Saudi comment on the attacks in Yemen.
Rashad al-Alimi, head of Yemen's Saudi-backed presidential council, on Sunday briefed U.S. Mission to Yemen Chargé d'Affaires Neil Hop on the developments, according to a statement.
He said the Houthis' attack put a responsibility on the international community, foremost among them the U.S., to "move beyond only containing the Houthi threat to addressing its sources and instruments".
The Yemeni military spokesperson said the attack on the port killed seven people.
The Yemeni transportation ministry said in a statement that the attack caused severe damage to the port's infrastructure.
"This marks yet another assault on a vital civilian facility that serves citizens and contributes to boosting commercial and maritime activity in the Red Sea," the ministry said.
Iran and its allied Shi'ite Muslim armed groups have been firing on Saudi Arabia and other Gulf states and blockading their energy shipments since the U.S. and Israel attacked the country on February 28 in a major escalation after years of regional tumult.
Talks have dragged on between the U.S. and Iran in pursuit of a peace deal, with access to the Strait of Hormuz, a major conduit for energy supplies, proving a major stumbling block.
Houthi attacks in the Red Sea have aggravated the disruptions of supplies to global markets.
Pakistan’s military, air force, prime minister’s office and information and foreign ministries did not respond to requests for comment and the Turkish government did not respond to a request on Sunday for comment on the implications of the refinery attack.
Turkish Foreign Minister Hakan Fidan said on Friday, as the alliance was announced, that it was not directed against Iran or any other country, but rather served as a general pledge to support the three allies' security.
Fidan said the allies would decide through consultations what degree, form, and format of support they would ask for in the event of an attack on any of them.