We report a highly sensitive method based on phase-stabilized swept source optical coherence elastography (PhS-SSOCE) to measure elastic wave propagation in soft tissues in vivo. The waves were introduced using a mechanical stimulus and were assessed using the phase response of the swept source optical coherence tomography signal. The technique was utilized to measure age-related changes in elastic flexural wave velocity and attenuation in mice cornea in vivo. Results demonstrate that the wave velocity increases with animal age, supporting previous observations that stiffness of mice cornea gradually increases with age. Our studies suggest that the PhS-SSOCE technique could potentially be used to obtain biomechanical properties of ocular tissues in vivo.