Due to the inevitable inter-study correlation between test sensitivity (Se) and test specificity (Sp), mostly because of threshold variability, hierarchical or bivariate random-effects models are widely used to perform a meta-analysis of diagnostic test accuracy studies. Conventionally, these models assume that the random-effects follow the bivariate normal distribution. However, the inference made using the well-established bivariate random-effects models, when outlying and influential studies are present, may lead to misleading conclusions, since outlying or influential studies can extremely influence parameter estimates due to their disproportional weight. Therefore, we developed a new robust bivariate random-effects model that accommodates outlying and influential observations and gives robust statistical inference by down-weighting the effect of outlying and influential studies. The marginal model and the Monte Carlo expectation-maximization algorithm for our proposed model have been derived. A simulation study has been carried out to validate the proposed method and compare it against the standard methods. Regardless of the parameters varied in our simulations, the proposed model produced robust point estimates of Se and Sp compared to the standard models. Moreover, our proposed model resulted in precise estimates as it yielded the narrowest confidence intervals. The proposed model also generated a similar point and interval estimates of Se and Sp as the standard models when there are no outlying and influential studies. Two published meta-analyses have also been used to illustrate the methods.