Theresa Riggi, 47, pleaded guilty to a charge of culpable homicide on the grounds of diminished responsibility. She was originally charged with murder.
Her eight-year-old twins Austin and Gianluca and their five-year-old sister Cecilia were found dead at their flat in Slateford Road on 4 August 2010.
They were discovered with multiple stab wounds after a suspected gas explosion at the property.
Sentencing of California-born Riggi, who denied recklessly causing the explosion, was deferred until 26 April.
The family had moved to the city from Skene in Aberdeenshire following the break-up of her marriage to the children’s father, Pasquale Riggi.
Mr Riggi said in a statement, read out by a spokesman from Victim Support Scotland, that the children would live in his heart forever.
He said: “The loss of my three beautiful children has been an overwhelming tragedy.
“Nothing can be said at this or any other time which can mitigate what has happened.
I and my extended family loved Austin, Luke and Cecilia dearly and they will live in our minds and hearts forever.
“I don’t know what lies ahead but wherever life takes me, my friends and family, it will be with the memory of a dreadful loss and with gratitude for all the good times spent with Austin, Luke and Cecilia.”
The High Court in Edinburgh heard that Riggi had been a protective mother who was involved in a custody battle with her estranged husband over access to the children.
