Fletcher, who has scored 11 of Sunderland's 33 top-flight goals this season, will miss the rest of the season after suffering ankle ligament damage while on international duty with Scotland on Friday.
Graham has yet to add to the seven goals he has scored for Swansea this season since moving to the Stadium of Light in January.
However, Johnson believes the 27-year-old is ready to make his mark at the club.
"I think Danny is a couple of games away from getting a good run," said Johnson, 25.
"It's important someone gets the goals with Fletch missing because we haven't scored that many or created many chances. When we get them, it's important to put them away.
"Fletch has proved himself this season. Without his goals, we would be in a lot worse position. He's going to be a big miss for us.
"I think it's important for the whole team to step up now and put the performances in we know we are capable of."