The baby girl is believed to have died March 2 or 3 after suffering from a flu-like virus that has affected many people in Baker Lake.
Six people have been medevaced out due to the virus.
The infant reportedly suffered with chronic diarrhoea for an extended period of time before dying.
Best said the review will be conducted by experienced medical professionals from outside the Nunavut health system.
"Looking into this matter, I don't like what I'm seeing," said Best.
"The investigation will look at the management or care of the deceased child since birth.
"We will also address other concerns that have been brought to our attention, as they pertain to the level of care patients are receiving in Baker Lake."
Mother files complaint
The baby's death comes about month after another child had to be rushed South.
Clara Noah filed a letter of concern on the way her child's care was handled in February.
Noah filed the letter with Tish Wilson, the patient service director in Rankin Inlet, on Feb. 12.
Noah's infant son, Nanurluk, has struggled with health problems during the first eight months of his life, including bouts with RSV and bronchiolitis.
Nanurluk also has an abnormal trachea.
The baby was suffering with a bout of croup and having difficulty breathing when Noah phoned the nurse on call.
"I phoned two or three times that evening and told her my son really had to be seen, but she just kept refusing," said Noah.
"When another nurse saw him the following morning, he was immediately medevaced to Churchill.
"The nurse kept asking me why I didn't phone the nurse on call the night before and I told them I did and she refused to see him."
Nanurluk was only kept in Churchill for 30 minutes before being sent to Winnipeg.
Noah said the child spent four days receiving oxygen in the pediatric intensive care unit in Winnipeg.
"If the local nurse-in-charge had seen him that evening and treated him with the ventilator, he probably wouldn't have had to go through all that.
"We have every right to seek, and receive, the medical attention we need.
"If that means having to be sent out for treatment, so be it. Just because we live up North shouldn't mean we can't access the same level of medical care every other Canadian receives."