We report an electrode-embedded on-chip platform technology for the precise determination of ultra-short (of the order of a few nanoseconds) relaxation times of dilute polymer solutions, by deploying time-alternating electrical voltages. Our methodology delves into the sensitive dependence of the contact line dynamics of a droplet of the polymer solution atop a hydrophobic interface in response to the actuation voltage, resulting in a non-trivial interplay between the time-evolving electrical, capillary, and viscous forces. This culminates into a time-decaying dynamic response that mimics the features of a damped oscillator having its 'stiffness' mapped with the polymeric content of the droplet. The observed electro-spreading characteristics of the droplet are thus shown to correlate explicitly with the relaxation time of the polymer solution, drawing analogies with a damped electro-mechanical oscillator. By corroborating well with the reported values of the relaxation times as obtained from more elaborate and sophisticated laboratory set-ups. Our findings provide perspectives for a unique and simple approach towards electrically-modulated on-chip-spectroscopy for deriving ultra-short relaxation times of a broad class of viscoelastic fluids that could not be realized thus far.