Abstarct: Saffman-Taylor instability is one of the interface instabilities that occurs when a less viscous fluid displaces a more viscous fluid and finger-like patterns appear on the interface of two immiscible fluids. In this numerical study, the Saffman-Taylor instability in immiscible viscoelastic-Newtonian displacement is presented for the first time. The simulation is performed in three dimension in a rectangular Hele-Shaw cell with two parallel plates and a small gap. Here, viscoelastic fluid is considered as the displacing fluid that displases the Newtonian fluid in the Hele-Shaw cell. The Giesekus model is used as the constitutive equation to describe the viscoelastic behavior. Also Oldroyd-B model as a quasi linear constitutive equation and Maxwell upper convected model are the special cases of Giesekus model that are considered to beter undrsrand the rheological properties of viscoelastic fluid. When the mobility factor parameter of Giesekus model takes the zero value, the Oldroyd-B model appears and when mobility factor be zero and the viscosity ratio takes one value the the Maxwel upper convected model appears. The volume of fluid method is applied to predict the formation of two phases and OpenFoam software is used for computational simulation. The main purpose of this study is the investigation of the effect of rheological properties of viscoelastic fluid and capillary forces on Saffman-Taylor instability in immiscible viscoelastic-Newtonian displacement. The results are reported in the form of dimensionless parameters. Investigated parameters are mobility ratio, viscosity ratio of viscoelastic model, mobility factor of Giesekus model, elasticity number and capillary number. Due to evaluate the effect of these parameters, sweep efficiency diagram and phase contours are presented for different values of dimensionless parameters. The effect of rheological properties and surface tension on the immiscible Saffman-Taylor instability are studied in detail. Results indicate that increasing the viscosity ratio, elasticity number, capillary number and extensional viscosity of viscoelastic phase stabilizes the flow field and enhances the sweep efficiency, while, increasing the mobility ratio and mobility factor causes the opposite effect and intensifies the instability. Also results show that the evaluated parameters has a sighnificiant effect on the mechanisms that appear in finger patterns like tip-splitting, coalescence, fading and shielding mechanisms. As a main consequence of this study, elastic property of viscoelastic fluid in the presence of capillary forces has a stabilize effect on the flow of Saffman-Taylor instability, while the shear-thinning prpperty increase the instability. The effects of this numerical study has a significiant effect in Enhance oil recovery processes of oil production and industry.