She told KUTV that she rushed home to Utah just in time to make it to the hospital before the boy was taken off life support.
The boy's father had also been on the phone from jail to say goodbye.
"We know that he loved Puniani so much," she said.
Maile told police he always slept with his Walther CCP handgun "fully loaded and safety off, within arm's reach", Fox News reported.
He has been charged with child abuse homicide, the Daily Mail reported.
Puniani was flown to Primary Children's hospital on May 25 local time and put on life support, but died three days later surrounded by family, including extended family who had reportedly flown out from New Zealand to support his parents.
In a post on Facebook, Puniani's grandmother and Maile's mother-in-law Tule Iloa wrote of her support despite the charges.
"I love you Tas and I know you love Puniani - this is a time of unity, love, forgiveness and one."
Salt Lake County District Attorney Sam Gill said police weighed several factors of the case carefully before charging Maile with homicide.
The way the young father stored his gun was a crucial factor.
"This is a little bit different than somebody who takes their weapon and puts it in a case up in the closet somewhere, hidden behind the clothes, and some child inadvertently finds it and shoots themselves," he said.
As well as the homicide charge, Maile has been charged with obstruction of justice and possession of a controlled substance owing to a blue bucket found by police containing 10 ounces of marijuana and a handgun magazine, the Daily Mail reported.
A neighbour said they saw Maile, with blood on his shirt, and his older son run to a dumpster and dispose of the bucket.
Officers also found a backpack with an empty gun case, believed to match the gun Maile was sleeping with, and two other handguns, court documents obtained by the Salt Lake Tribute said.
Maile has been released on bond.
Puniani's mother was not home at the time and police said they found no furniture in the room Maile slept in with his two boys, indicating the trio may have been sleeping on the floor.
In a loving obituary, little Puniani was described as the "light of his family".
"He absolutely kept the family together with his love and welcoming attitude. He was full of life and was such a character," it read.
"He greeted every person he met with his contagious smile and upbeat energy. The memories he has left with his family and friends will forever be cherished and held dearly."
A Gofundme page set up to pay for his funeral had attracted more than US$3000 in donations in 10 days.
