Solid-state nanopores can act as single-molecule sensors and could potentially be used to rapidly sequence DNA molecules. However, nanopores are typically fabricated in insulating membranes that are as thick as 15 bases, which makes it difficult for the devices to read individual bases. Graphene is only 0.335 nm thick (equivalent to the spacing between two bases in a DNA chain) and could therefore provide a suitable membrane for sequencing applications. Here, we show that a solid-state nanopore can be integrated with a graphene nanoribbon transistor to create a sensor for DNA translocation. As DNA molecules move through the pore, the device can simultaneously measure drops in ionic current and changes in local voltage in the transistor, which can both be used to detect the molecules. We examine the correlation between these two signals and use the ionic current measurements as a real-time control of the graphene-based sensing device.