The first opportunity fell to the home side, who were energised by a packed crowd at Ashton Gate, as Mehmeti received possession and drove forward before shooting narrowly wide.
The 24-year-old attacking midfielder continued to look dangerous throughout the opening stages, racing in behind the Charlton back line but defender Kayne Ramsay was on hand to provide a super tackle and prevent a shot on goal.
Bristol City dominated the possession and shots in the first half but neither side was able to manage a shot on target in the opening 45 minutes.
Charlton started the second half well and came close as Rob Apter fired a dangerous effort but Robins keeper Radek Vítek got down well to make the stop.
Mehmeti was agonisingly close to opening the scoring just before the hour mark as he thundered a shot off the post.
Vítek produced a fine stop moments later to keep the score level, clawing the ball onto the post to prevent an own goal as George Tanner nearly put it into his own net.
As the game went into seven minutes of added time, Charlton summer signing Harvey Knibbs had a close-range chance smothered by Vítek.