We investigate the effect of the Dzyaloshinskii-Moriya interaction (DMI) on magnetic domain nucleation in a ferromagnetic thin film with perpendicular magnetic anisotropy. We propose an extended droplet model to determine the nucleation field as a function of the in-plane field. The model can explain the experimentally observed nucleation in a Co/Ni microstrip with an interfacial DMI. The results are also reproduced by micromagnetic simulation based on the string model. The electrical measurement method proposed here can be widely used to quantitatively determine the DMI energy density.