Cultural sustainability is increasingly being perceived as a fourth dimension of sustainable development. So far, some studies have debated the way in which cultural sustainability can lead to economic, social, and environmental benefits, while others have highlighted how the classic pillars of sustainability can help museums to achieve their core cultural mission. However, empirical studies regarding cultural sustainability in museums are scarce. Thus, one of the aims of our research was to fill this gap by developing several econometric models that explain the influence of heritage exposure; environmental behavior; openness to the public; and effectiveness and performance in collecting, preserving, and researching the cultural heritage. A second aim was to advance the current knowledge in this field by creating an integrated frame that explains the interconnections between different variables that help museums become sustainable, as well as the place and role of cultural sustainability within the overall framework of sustainable development. To achieve these goals, an in-depth analysis of the literature was followed by a survey of 86 Romanian museums. The results show that the ability of museums to reach cultural sustainability is influenced by components of their social and economic performance, while environmental behavior proved to be insignificant.