The facility is run by Oceania and is being supported by Public Health and Auckland DHB.
The testing was being undertaken with residents and staff as part of the outbreak management plan, MOH said.
One resident at the rest home was transported to Auckland Hospital on New Year's Eve and confirmed as Covid-19 positive.
"We all know Covid-19 is in the community. A person in the community has brought Covid into our community at Everil Orr," said Brent Pattison, chief executive of Oceania Healthcare.
"We had an elderly resident with serious health conditions transferred to hospital, who subsequently tested positive for Covid-19.
"All our residents and staff have been tested by the Public Health Unit. We have five residents, including the one in hospital and one staff member that have tested positive for Covid-19 and are isolating."
Pattison said all its residents and staff are double vaccinated and have recently received their boosters.
"We have planned for such an event and our staff continue to provide quality care to our residents," Pattison said.
Last year, two West Auckland rest homes had also been hit with residents and staff testing positive for Covid-19.
More than 20 residents and four staff members were diagnosed with the virus at Edmonton Meadows Care Home in Henderson, and two residents at Rosaria Rest Home in Avondale also tested positive.
One woman at Edmonton Meadows Shanti Kumari, 87, tested positive for Covid-19 later died but the ministry said she had died from an unrelated cause.
