The couple's children said Tuesday that they died "peacefully and under circumstances of their own choosing" on Friday at a Zurich clinic run by the group Dignitas.
"After 54 happy years together, they decided to end their own lives rather than continue to struggle with serious health problems," said a statement from the couple's son and daughter, Caractacus and Boudicca.
The statement said Downes, who became Sir Edward when he was knighted by Queen Elizabeth II in 1991, had "a long, vigorous and distinguished career," but in recent years had become almost blind and nearly deaf.
His wife Joan, a former dancer, choreographer and television producer, had devoted years to working as his assistant. Downes' manager, Jonathan Groves, said Joan Downes had been diagnosed with terminal cancer.
"Lady Downes was terminally ill, Sir Edward wasn't," he said. "It was a decision they both reached. Sir Edward would have survived her death but he decided he didn't want to. He didn't want to go on living without her."
Groves said he was shocked by the couple's deaths but called their decision "typically brave and courageous."
He said Edward Downes was "a man of extraordinary self-determination in his absolutely uncompromising pursuit of achieving the highest standards musically," who had struggled in old age as his eyesight and hearing failed.
London's Metropolitan Police force said it had been notified of the deaths, and was investigating.
The double suicide is the latest in a series of high-profile cases that have spurred calls for a legal change in Britain, where assisted suicide and euthanasia are banned.
More than 100 Britons have died in Swiss clinics run by Dignitas. The organization was established in 1998, and takes advantage of the country's liberal laws on assisted suicide, which suggest that a person can be prosecuted only if they are acting out of self interest.
Roughly 100 foreigners most of them terminally ill come to Switzerland each year to take advantage of the country's liberal laws on assisted suicide. Some are healthy except for a disability or severe mental disorder. Typically they go to a room run by Dignitas, which provides them with a lethal drink of barbiturates. In five minutes they fall asleep and never wake up.
Other countries, including the Netherlands and Belgium, and the states of Oregon and Washington in the United States, allow the incurably sick to obtain help from a doctor to hasten their death.
But only Switzerland, in a law dating back to 1942, permits foreigners to come and kill themselves. Other organizations provide such services for Swiss residents, but Dignitas is the main organization for foreigners.
Critics accuse Dignitas of promoting "suicide tourism."
Dignitas says it is meeting the needs of its members. It says it charges 10,000 Swiss francs ($9,200) for its services, which include taking care of legal formalities and arranging consultations with a doctor willing to prescribe the barbiturates.
British law is clearly against assisted suicide, but enforcement has been somewhat lax.
British courts have been reluctant in recent years to convict people who help loved ones travel to clinics abroad to end their lives. No relative or friend of any of the Britons who have died in Dignitas clinics has been prosecuted.
But parliamentary efforts to change the rules have all been defeated most recently last week, when Parliament's upper chamber, the House of Lords, voted down an amendment that would have relaxed the prohibition on assisted dying.
Sarah Wootton, chief executive of campaign group Dignity in Dying, said the Downes' deaths showed the need to regulate assisted suicide.
Edward Downes is one of the most prominent Britons to have traveled to Switzerland because of its open attitude toward foreigners seeking to end their lives.
He was born in 1924 in Birmingham in central England. He studied at Birmingham University, the Royal College of Music and under German conductor Hermann Scherchen.
In 1952 he joined London's Royal Opera House as a junior staffer
his first job was prompting soprano Maria Callas. He made his debut as a conductor with the company the following year and went on to become associate music director. Throughout his life he retained close ties to the Royal Opera, conducting almost 1,000 performances of 49 different operas there over more than 50 years.
He also had a decades-long association with the BBC Philharmonic Orchestra, where he became principal conductor and later conductor emeritus.
Groves said that during Downes' 70s and 80s, he slowly went blind, "which was a huge problem professionally not being able to read scores. That had become much worse in recent years.
"His hearing was also deteriorating, which was another problem."
The couple is survived by their children and family said there would be no funeral.