Young's modulus, E(H), in as-quenched $Fe_{81}B_{13.5}Si_{3.5}C_2$ metallic glass was measured using impedance resonance method as a function of magnetic field under applied tensile stress. E(H) with magnetic field under zero applied stress decreses at low field about 1 Oe and then increased to 10 Oe, over 10 Oe saturated. Under zero applied stress the minimum value and saturation value of E(H) was 123 GPa, 170 GPa respectively and maximum of $\Delta$E was 0.38. As the applied tensile stress incresed, Young's modulus varied with magnetic field similar to that of under zero stress, but $\Delta$E decreased. The saturation magnetostriction coefficient, $\lambda$s was otained from the saturating field of Young's modulus, $H_n$ and applied stress; $\lambda$s was $29.7\times10^{-6}$ which was the same value obtained using small angle variation method. The variation of Young's modulus with magnetic field was calculated using single domain model under the assumption that the Young's modulus is proportional to the transverse volume fraction of domain, v$\perp$, and the v$\perp$ was decresed with applied tensile stress. The computer simulation of E(H) under several applied tensile stress agrees well with experimental results suggesting that the variation of E(H) is due to the rotation of the transverse domain toward applied magnetic field direction.