The first strike came around 11 pm yesterday and hit a building housing the facility's administration offices, according to Hassan Boucenine, who spoke to The Associated Press by telephone from the southern port city of Aden.
No one was inside at the time, he said, adding that by the time a second strike targeted the main nearby building about 10 minutes later, its occupants, some 12 staff and patients, had been evacuated.
A Saudi-led, US-backed coalition has been launching airstrikes against Yemen's Shiite rebels, also known as Houthis, and their allies since March.
Saada, the Houthis stronghold, has faced a particularly intense bombardment.
The United Nations said the facility was the 39th health center hit since the violence escalated in March, adding that critical shortages of fuel, medication, electricity and water could mean many more will close. Amnesty International said the strike may amount to a war crime and called for an independent investigation.