The president of Rapa Nui's indigenous Ma'u Henua community said the case was confirmed today following a first case two days ago.
Camilo Rapu said it was not known how the first person, a Chilean woman living on the island, contracted the illness as she had not been overseas or believed to be in contact with an infected person.
Mr Rapu said none of the woman's family was showing any symptoms.
Nothing was known yet about the second case as the community was in isolation.
The lockdown has caused frustration among many islanders over how long it took Chilean authorities to act after cases on the mainland.
Rapa Nui's mayor, Pedro Edmunds Paoa, who had called for the island's immediate isolation has been vocal in his criticism of Santiago's handling.
The Guardian reported Latam, which provides the island's only air link, suspended flights on Tuesday after evacuating all tourists.
Like many Pacific islands, Rapa Nui has no intensive care or Covid-19 testing facility and relies on mainland Chile for support.
An indigenous tour operator on the island said he understood more cases were being investigated.
Sebastián Paoa Águila said central government needed to extend the quarantine of the island indefinitely.
Mr Paoa Águila said Santiago was not being very forthcoming with information and it was causing fear.
"In the case of Easter Island now, we have two cases confirmed. The second one is today but there [may] be another three cases.
"They still don't know but they send to Santiago because the island is really really small and we don't have a lot of resources."
The island relied upon tourism for income but they could get by until the pandemic was contained, he said.