In this paper, a generalization of the Gaussian quasi likelihood ratio test (GQLRT) for simple hypotheses is developed. The proposed generalization, called measure-transformed GQLRT (MT-GQLRT), selects a Gaussian probability model that best empirically fits a transformed probability measure of the data. By judicious choice of the transform we show that, unlike the GQLRT, the proposed test can gain sensitivity to higher-order statistical moments and resilience to outliers leading to significant mitigation of the model mismatch effect on the decision performance. Under some mild regularity conditions we show that the proposed test statistic is asymptotically normal. A data driven procedure for optimal selection of the measure transformation parameters is developed that maximizes an empirical estimate of the asymptotic power given a fixed empirical asymptotic size. The MT-GQLRT is applied to signal classification in a simulation example that illustrates its sensitivity to higher-order statistical moments and resilience to outliers.