Shane (17), who was named after the former Pogues frontman, had been missing since Thursday and his remains were found in Bray, Co. Wicklow on Friday.
In an online post McGowan described Shane as “beautiful boy” and said he loved him.
“Sinead you have always been there for me and for so many people, in this world you have been a comfort and you have been a soul who is not afraid to feel the pain of the suffering and you have always tried to heal and to help.
“I pray and Victoria prays that you can be comforted now and find strength and healing and peace in your own sorrow and loss, he was a beautiful boy and I loved him and I pray that he can be at peace and also be able to stay connected with you."
Sinead described Shane, who struggled with mental health issues, as the light of her life.
"My beautiful son, Nevi’im Nesta Ali Shane O’Connor, the very light of my life, decided to end his earthly struggle today and is now with God. May he rest in peace and may no one follow his example. My baby. I love you so much. Please be at peace."
Sinead said earlier this week that her son had gone missing from Tallaght Hospital.
Socialist Party TD Ruth Coppinger said more funding needed to be given to deal with mental health in Ireland.
“How incredibly sad and horrific for Sinead O'Connor and Donal Lunny to experience the loss of their son this way. Also, very strong to not hide the cause of death. The mental health crisis is real and disgracefully neglected: massive investment & resources needed by the state. RIP.”
Psychologist and broadcaster Emma Kenny described the news as “tragic beyond belief” and said young people are being failed in society.
She added: “We are failing a whole generation; no child should feel death is a better option. As a society we need to ask ourselves why and we need to do better.”
Children’s author Philip Ardagh said his heart goes out to Sinead at such a tragic time.
“I have a huge admiration for Sinead O’Connor who spoke out publicly against the Catholic Church many, many years before most dared to, and was shunned and ridiculed as a result. And now she has lost her 17-year-old son, Shane. My heart goes out to her.”
Labour Party Senator Annie Hoey also offered her sympathy.
“Oh my goodness I am so sorry Sinead. Sending light, strength and love to you and all who loved him.”
Best-selling author Shelia O’Flanagan said: “Oh, Sinead. So very, very sorry for your loss. Love to all your family. RIP.”
Businesswoman and television personality Norah Casey wrote to Sinead saying: “Sending you strength and heartfelt sympathies
“No words can ease your broken heart - will light candles for your lovely son Shane - may he rest in peace.”
Shane’s father was Donal Lunny, one of Ireland's most admired traditional music stars and best known for his roles in Planxty, The Bothy Band, Moving Hearts.