The incident happened in the densely populated West Point township on Saturday evening.
At least 20 patients who were being monitored for signs of the illness have left the centre.
Officials said blood-stained bedding looted from the centre posed a serious infection risk.
The protesters were unhappy that patients were being brought in from other parts of the capital, the assistant health minister said.
Other reports suggested the protesters believed Ebola was a hoax and wanted to force the quarantine centre to close.
The centre was set up to observe suspected Ebola patients and then transfer them to a main treatment centre if they prove positive, assistant health minister Tolbert Nyenswah told the BBC.
It is not known if those at the centre were infected with the virus, though one report suggested they had proved positive.
A senior police officer said blood-stained mattresses, beddings and medical equipment were taken from the centre.
"This is one of the stupidest things I have ever seen in my life", he said.
He said the looting spree could threaten to spread the virus to the whole of the West Point area.
Described as a 'slum', there are an estimated 50,000 people in the West Point neighbourhood.
The Ebola epidemic began in Guinea in February and has since spread to Liberia, Sierra Leone and Nigeria.
On Friday, the death toll rose to 1,145 after the WHO said 76 new deaths had been reported in the two days to 13 August. There have been 2,127 cases reported in total.
The attack at the Monrovia centre is seen as a major setback in the struggle to halt the outbreak, says the BBC's Will Ross, reporting from Lagos.
Health experts say that the key to ending the Ebola outbreak is to stop it spreading in Liberia, where ignorance about the virus is high and many people are reluctant to cooperate with medical staff.
* Symptoms include high fever, bleeding and central nervous system damage.
* Fatality rate can reach 90% - but the current outbreak is about 55%.
* Incubation period is two to 21 days.
* There is no vaccine or cure.
* Supportive care such as rehydrating patients who have diarrhoea and vomiting can help recovery.
*Fruit bats are considered to be virus' natural host.