This paper presents a hybrid optimization approach for the determination of the optimum laser cutting process parameters which minimize the kerf width, kerf taper, and kerf deviation together during pulsed Nd:YAG laser cutting of a thin sheet of nickel-based superalloy SUPERNI 718 (an equivalent grade to Inconel 718). A hybrid approach of Taguchi methodology and grey relational analysis has been applied to achieve better cut qualities within existing resources. The input process parameters considered are oxygen pressure, pulse width, pulse frequency, and cutting speed. A higher resolution based L 27 orthogonal array has been used for conducting the experiments for both straight and curved cut profiles. The designed experimental results are used in grey relational analysis and the weights of the quality characteristics are determined by employing the entropy measurement method. The significant parameters were obtained by performing analysis of variance (ANOVA). The optimized parameters for straight and curved laser cut profiles have been compared. On the basis of optimization results it has been found that the optimal parameter level suggested for straight cut profiles are not valid for curved cut profiles. The application of the hybrid approach for straight cuts has reduced K t and K d by 52.37% and 17%, respectively. For curved cuts the approach has reduced K w and K t by 8.45% and 44.44%, respectively. The results have also been verified by running confirmation tests.