The increasing share of natural gas-fired power plants in the electricity generation mix has tightened the existing link between power and gas systems. This may bring technical and economical risks, e.g., congestions and peak prices, which mainly stem from the non-coordinated operation of both systems and the failure to properly anticipate the multivariate and correlated demand uncertainty. In this paper, we propose a Wasserstein metric-based distributionally robust optimization framework for the joint power and gas dispatch to mitigate these risks. The distributionally robust decision-making approach strives to ensure risk-hedging dispatch solutions, by considering a set of potential distributions, the so-called ambiguity set. We go beyond the state of the art by considering a realistic dependence structure among the power and gas demands uncertainty. We show the benefits of the proposed approach via a numerical study based on the Belgian networks.