The Glovers took the lead through Gavin Tomlin in the 15th minute. The home defence failed to clear Darren Way's corner and keeper Ben Williams could only parry Marc Bircham's low drive straight into the path of Tomlin who pounced to ram the ball home from six yards for his third goal of the season.
Carlisle drew level a minute into first-half stoppage time when Danny Livesey nodded the ball across goal and Graham's control enabled him to turn and slide a shot just inside Asmir Begovic's left-hand post from the edge of the six yard box.
In the second-half Cleveland Taylor's cross created a chance for Danny Carlton, and when his header was blocked by Begovic, the effervescent Graham was alert to the opportunity and pounced to slot home home the rebound.
A rash foul on Taylor by Lee Peltier gave Graham the chance to complete the hat-trick from the penalty spot after 69 minutes, and he made no mistake by side-footing past Begovic and into the bottom-right hand corner for his eighth goal of the season.
Substitute Gary Madine rounded off the scoring in the 90th minute with his first league goal as he slid home Taylor's cross from the left.