We report the observation of the rare charm decay $D^0\rightarrow K^-\pi^+e^+e^-$ , based on $468$ fb$^{-1}$ of $e^+e^-$ annihilation data collected at or close to the center-of-mass energy of the \FourS\ resonance with the BaBar detector at the SLAC National Accelerator Laboratory. We find the branching fraction in the invariant mass range $0.675< m(e^+e^-) < 0.875$ GeV/c$^{2}$ of the electron-positron pair to be $\mathcal{B}(D^0\rightarrow K^-\pi^+e^+e^-) = (4.0 \pm 0.5 \pm 0.2 \pm 0.1) \times 10^{-6}$, where the first uncertainty is statistical, the second systematic, and the third due to the uncertainty in the branching fraction of the decay $D^0\rightarrow K^-\pi^+\pi^+\pi^-$ used as a normalization mode. The significance of the observation corresponds to 9.7 standard deviations including systematic uncertainties. This result is consistent with the recently reported $D^0\rightarrow K^-\pi^+\mu^+\mu^-$ branching fraction, measured in the same invariant mass range, and with the value expected in the Standard Model. In a set of regions of \mee\ where long-distance effects are potentially small, we determine a 90\% confidence level (CL) upper limit on the branching fraction $\mathcal{B}(D^0\rightarrow K^-\pi^+e^+e^-) < 3.1\times 10^{-6}$.