As the first endeavor, we have analyzed the pulsatile flow of Oldroyd-B viscoelastic fluid where the combined effects of fluid elasticity and pulsation parameters on the flow characteristics are numerically studied at a low Reynolds number. Computations are performed using a finite volume-based open-source solver OpenFOAM™ by appending the log-conformation tensor approach to stabilize the numerical solution at high Deborah number. Significant flow velocity enhancement is achieved by increasing the viscoelastic behavior of the fluid. High-velocity gradient zones and high polymeric stress regions are observed near the channel wall. The magnitude of axial velocity attenuates with increasing pulsation amplitude or pulsation frequency, and the extent of this attenuation is highly dependent on the Deborah number or the retardation ratio. This work finds application in the transport of polymeric solutions, extrusion, and injection molding of polymer melts in several process industries.