Symmetrically coupled oscillators represent a limiting case for studying the dynamics of natural systems. Therefore, we here investigate the effect of coupling asymmetry on delay-induced oscillation death (OD) in coupled nonlinear oscillators. It is found that the asymmetrical coupling substantially enlarges the domain of the OD island in the parameter space. Specifically, when the intensity of asymmetry is enhanced by turning down the value of the coupling asymmetry parameter α, the OD island gradually expands along two directions of both the coupling delay and the coupling strength. The expansion behavior of the OD region is well characterized by a power law scaling, R=α(γ) with γ≈-1.19. The minimum value of the intrinsic frequency, for which OD is possible, monotonically decreases with decreasing α and saturates around a constant value in the limit of α→0. The generality of the conducive effect of coupling asymmetry is confirmed in a numerical study of two delay-coupled chaotic Rössler oscillators. Our findings shed an improved light on the understanding of dynamics in asymmetrically delay-coupled systems.