Gough, a product of Charlton's youth system, has been released by the Addicks following his first year as a professional.
The 18-year-old, who stands at a towering 6ft 7in, has previously enjoyed non-league loan spells with St Albans, Salisbury, Eastbourne and Chelmsford.
"It's going to be a wrench leaving Charlton, but I think change is good," Gough told Bristol Rovers official website. "Perhaps I was getting a bit too comfortable there and now I need to get out of my comfort zone and prepare for a new challenge.
"My agent organised a trial for me with Rovers, so I came down. The gaffer said I did well and I played in an in-house game and it's gone from there.
"I still have a great deal of work to do, but hopefully I'll be pushing Scott Bevan for a first-team place next season and get my opportunity."