Birnbaum-Saunders models have been widely used to describe data following positive-skew distributions. In this article, we introduce a bivariate Birnbaum-Saunders distribution which has the mean as one of its parameters, allowing us to mimic the standard parameterization of the normal distribution, but in an asymmetric framework. We derive some properties of the mean-based bivariate Birnbaum-Saunders distribution useful for statistical and reliability analyses. Maximum likelihood and modified moment estimations of the model parameters and associated inference are considered. A simulation study is conducted to evaluate the performance of the corresponding estimators and coverage probabilities of confidence intervals are also discussed. Finally, a real-world data analysis is carried out for illustrating the potential of the proposed model.