The 25-year-old was said to have been struggling after burying his father in Ireland last week.
Cloud played drug dealer Fezco in 17 episodes of the hit HBO drama, alongside the likes of Zendaya and Hunter Schafer.
A family statement said: “It is with the heaviest heart that we had to say goodbye to an incredible human today.
“As an artist, a friend, a brother and a son, Angus was special to all of us in so many ways.
“Last week he buried his father and intensely struggled with this loss.
“The only comfort we have is knowing Angus is now reunited with his dad, who was his best friend.”
The statement added: “Angus was open about his battle with mental health and we hope that his passing can be a reminder to others that they are not alone and should not fight this on their own in silence.
“We hope the world remembers him for his humour, laughter and love for everyone. We ask for privacy at this time as we are still processing this devastating loss.”
US site TMZ reported that Cloud’s mother had called 911 on Monday about a “possible overdose”.
Emergency services told NBC News they were called at about 11.30am but the “patient was determined to already be deceased” when they arrived.
They said the cause of death was currently unknown.
