We demonstrate the potential of perylene bisimide (PBI) for label-free sensing of organic molecules by investigating the change in electronic properties of five symmetric and asymmetric PBI derivatives, which share a common backbone, but are functionalised with various bay-area substituents. Density functional theory was combined with a Greens function scattering approach to compute the electrical conductance of each molecule attached to two gold electrodes by pyridyl anchor groups. We studied the change in their conductance in response to the binding of three analytes, namely TNT, BEDT-TTF and TCNE, and found that the five different responses provided a unique fingerprint for the discriminating sensing of each analyte. This ability to sense and discriminate was a direct consequence of the extended π system of the PBI backbone, which strongly binds the analytes, combined with the different charge distribution of the five PBI derivatives, which leads to a unique electrical response to analyte binding.