Both bilevel and robust optimization are established fields of mathematical optimization and operations research. However, only until recently, the similarities in their mathematical structure has neither been studied theoretically nor exploited computationally. Based on the recent results by Goerigk et al. (2025), this paper is the first one that reformulates a given strictly robust optimization problem with a decision-dependent uncertainty set as an equivalent bilevel optimization problem and then uses solution techniques from the latter field to solve the robust problem at hand. If the uncertainty set can be dualized, the respective bilevel techniques to obtain a single-level reformulation are very similar compared with the classic dualization techniques used in robust optimization but lead to larger single-level problems to be solved. Our numerical study shows that this leads to larger computation times but may also slightly improve the dual bound. For the more challenging case of decision-dependent uncertainty sets represented by mixed-integer linear models we cannot apply standard dualization techniques. Thus, we compare the presented bilevel approach with the only available method from the literature, which is based on quantified mixed-integer linear programs. Our numerical results indicate that, for the problem class of decision-dependent robust optimization problems, the bilevel approach performs better in terms of computation times.