Brayford has arrived from Derby County, but he has found himself on the bench for City's first two Premier League matches, and Connolly is aware that he must remain consistent to avoid being replaced.
The 25-year-old told Wales Online: "To be fair, I put pressure on myself to perform in every game. I know John Brayford has come in and he's a very good player so I know that I have to perform in every game if I am going to keep my place.
"It's fantastic to be a Premier League player. I've waited for this chance having been promoted a few times so to finally get the chance is great.
"Now I want to play every game and must make sure that I'm performing at a good enough standard to stay in the team."
Connolly made 34 appearances as Cardiff won the Championship last season.