In this paper, the well-known Saffman-Taylor instability of an immiscible quasilinear viscoelastic-Newtonian displacement in a Hele-Shaw cell is studied numerically for the first time. The volume of fluid method is applied to predict the formation of two phases. Here, a quasilinear viscoelastic fluid is considered as the displacing fluid and a Newtonian fluid as the displaced fluid. The Oldroyd-B constitutive equation, which is physically useful for Boger liquids, is considered for the viscoelastic phase. The effect of dimensionless parameters, consisting of the viscosity ratio, the viscosity ratio of viscoelastic fluid, the elasticity number, and the capillary number on Saffinan-Taylor instability are studied in detail. The results illustrate that increasing the capillary number, elasticity number, and the viscosity ratio of viscoelastic fluid stabilizes the displacement, while enhancing the viscosity ratio has a destabilizing effect on the displacement. As a main result, it is found that the elasticity of the displacing fluid has a stabilizing effect on the flow field in the presence of capillary forces, which can be attributed to enhancing the extensional viscosity that resists against the stretching of the fingers.