We study the spreading dynamics of a sphere-shaped elastic non-Newtonian liquid drop on a spherical substrate in the capillary-driven regime. We use the simplified Phan-Thien-Tanner model to represent the rheology of the elastic non-Newtonian drop. We consider the drop to be a crater on a flat substrate to calculate the viscous dissipation near the contact line. Following the approach compatible with the capillary-viscous force balance, we establish the evolution equation for describing the temporal evolution of the contact line during spreading. We show that the contact line velocity obtained from the theoretical calculation matches well with our experimental observations. Also, as confirmed by the present experimental observations, our analysis deems efficient to capture the phenomenon during the late stage of spreading for which the effect of line tension becomes dominant. An increment in the viscoelastic parameter of the fluid increases the viscous dissipation effect at the contact line. It is seen that the higher dissipation effect leads to an enhancement in the wetting time of the drop on the spherical substrate. Also, we have shown that the elastic nature of the fluid leads to an increment in the dynamic contact angle at any temporal instant as compared to its Newtonian counterpart. Finally, we unveil that the phenomenon of the increasing contact angle results in the time required for the complete wetting of drop, which becomes higher with increasing viscoelasticity of the fluid. This article will fill a gap still affecting the existing literature because of the unavailability of experimental investigations of the spreading of the elastic non-Newtonian drop on a spherical substrate.