Singularities in the spectra of open systems, known as exceptional points (EPs), have been shown to exhibit nontrivial topological properties and enhanced sensitivities. Here, we propose a novel approach to realize the EPs in a plasmon-exciton hybrid system and explore their applications in enhanced nanoscale sensing technology. We consider a plasmon-exciton system composed of a gold nanorod and a monolayer WSe2. By controlling the geometric parameters of the nano-hybrid system, we obtain simultaneous coalescence of the resonance frequencies and loss rates of the hybrid system, which is a unique feature of EPs. Numerical simulations show its application in enhanced nanoscale sensing for environmental refractive indices. Our work opens the way to a new class of sensors based on EP-enhanced sensing, with intrinsic nanoscale sensitivity due to the sub-diffraction-limit size of the plasmon-exciton nano-hybrid system.