Introduced exotic species can dominate communities and replace native species that should be better adapted to their local environment, a paradox that is usually explained by the absence of natural enemies and by habitat alteration resulting from anthropogenic disturbance. Additionally, introduced species can enhance their invasion success and impact on native species by modifying selection pressures in their new environment through ecosystem engineering. We analyse a simple dynamic model of indirect competition for habitat between a non-engineering resident species and an engineering exotic species. The conditions for invasion and competitive exclusion of the resident by the exotic species and the range of dynamic outcomes suggested by the model are determined by the form of density dependence. We give simple criteria for the success of the invading species on dimensionless quantities involving rates of ecosystem engineering and of habitat degradation. The model's predictions offer an additional explanation for a range of invasion dynamics reported in the literature, including lag times between introduction and establishment. One intriguing result is that a series of failed invasions may successively reduce environmental resistance to subsequent invasion, through a cumulative effect of habitat transformation. More work is needed to determine the frequency and conditions in which engineering is required for successful establishment, and whether highly-successful (or high-impact) invaders are more likely to possess ecosystem engineering traits.