Almost 250 people have died of the disease this month alone, with hundreds of suspected cases being reported every day, it says.
The World Health Organization said the water-borne illness is spreading at an alarming rate in the war-torn country.
Sanaa has been the worst affected area, with a state of emergency declared in the rebel-held capital last week.
Save the Children fears thousands of people could die of the easily treatable disease, and said more than two million malnourished children are particularly at risk.
Cholera is a water-borne disease that is transmitted through contaminated water and food.
Symptoms include acute diarrhoea and vomiting. People ill with cholera can become very sick and, when it is left untreated, death can occur within hours.
Charity Médecins Sans Frontières (MSF), which has set up cholera treatment centres in the country, also fears the outbreak will spiral out of control.
“Before the outbreak, the health system was already overstretched and people’s health needs were already huge,” says Ghassan Abou Chaar, MSF’s head of mission in Yemen.
“To bring the outbreak under control, it won’t be enough simply to treat those people who reach medical facilities. We also need to address the source of the disease, by improving water and sanitation and working in communities to prevent new cases.”
The WHO says fewer than 45% of health facilities are fully functioning in Yemen, with almost 300 damaged or destroyed in fighting between forces loyal to President Abdrabbuh Mansour Hadi – who is backed by a Saudi-led multinational coalition – and those allied to the Houthi rebel movement.
Save the Children said all sides in the conflict should end restrictions on the import of aid immediately.
More than eight million people lack access to drinking water and sanitation.
More than 8,000 people – mostly civilians – have been killed and close to 44,500 others injured since the conflict in Yemen escalated in March 2015, according to the UN.
The fighting has also left 18.8 million people in need of humanitarian assistance.