Sister Clare Theresa Crockett, 33, died when a stairwell collapsed in the school she was working at in Playa Prieta.
More than 200 people died in Ecuador when a powerful earthquake hit the south American country on Saturday.
Thousands of people were injured in collapsing buildings, roads and bridges.
Sister Clare, from Derry city, was a nun in the Home of the Mother order and had been teaching children in a rural part of the country, including how to play the guitar.
"She was a superstar. Everybody loved her," her cousin Emmet Doyle said.
A number of young women died alongside the nun.
Her family said: "On Sunday 17th April, we lost our daughter, sister and aunt Sister Clare Theresa Crockett as a result of the earthquake in Ecuador.
"She was situated in a school in Playa Prieta with the Home of the Mother order. At this difficult time we would ask for privacy."
Her family believes the missionary nun was trying to lead girls out of the school after the earthquake hit when the stairwell collapsed.
"She has been all over the world, working as a nun for 15 years. She was only 33," Mr Doyle said.
"She had been teaching children that day, she had been teaching them how to play the guitar.
"She was trying to get them down the stairs and the staircase collapsed. We knew she was trapped but information has been slow to come out.
"She died as she lived, helping others."
Sr Clare worked in Spain, the US and other parts of the world before going to Ecuador.
The Servants of the Home of the Mother said six of their members died in the tragedy.
The five others were postulants, one of the early stages in formation before becoming a nun.