Numerical simulations of fluid-structure interaction typically require vast computational resources. Finite-element techniques employing goal-oriented hp-adaptation strategies could offer a substantial improvement in the efficiency of such simulations. These strategies rely on dual-based a-posteriori error estimates for quantities of interest. However, the free-boundary character of fluid-structure-interaction problems forms a fundamental complication, as it yields the underlying domain unknown a-priori. Instead, the domain comprises part of the solution. Consequently, the well-established generic framework for dual-based error estimation is not applicable. In this work we develop a framework for dual-based a-posteriori error estimation for free-boundary problems such as fluid-structure interaction. The framework is based on the embedded-domain approach and an extension operator which enables the comparison of approximate solutions on distinct domains. Given an approximate fluid and structure solution, we present a dual problem on the corresponding approximate fluid domain. Finally, we employ the dual solution to present an exact error representation formula.