"Allies" revolves around a televised 1999 meeting of the two late leaders when Pinochet was being held under house arrest in Britain.
An anonymous Argentine soldier conscripted in the 1982 war over the Falkland Islands, known as the Las Malvinas in Argentina, provides a counterpoint alongside two other – a Pinochet aide and Thatcher's nurse.
At the time of the meeting, at which Thatcher famously thanked her old friend and Falklands ally for "bringing democracy to Chile", Pinochet was fighting a legal battle to avoid extradition to Spain on human rights abuse charges.
Thatcher remained a staunch defender of Pinochet for the rest of her life following his support for Britain during the Falklands conflict with Argentina.
Writer Esteban Buch said the opera was openly political and certainly not an invitation "to admire Pinochet or Thatcher".
"The Malvinas War is the experience of my generation of Argentines. Through Thatcher and Pinochet and the Malvinas War I looked back on my past and the past of my generation," he said in production notes.
"I absolutely assume a political, committed message, which is clear," he said, adding that he wanted to trigger in the audience "something more than the simple affirmation (that) we are against these monsters".
Pinochet, who ruled Chile with an iron fist from 1973-1990, was linked to thousands of cases of torture, abduction and death.
He died aged 91 in 2006 after a heart attack without ever standing trial on any charge. His lawyers successfully argued that "mild dementia" prevented him from defending himself.
Thatcher died in April aged 87 following a stroke and a long battle with dementia.
"Allies" will also be performed in Rome and Strasbourg in October and at Saint-Quentin-en-Yvelines near Paris in January.