Sensorimotor control in vertebrates relies on internal models. When extending an arm to reach for an object, the brain uses predictive models of both limb dynamics and target properties. Whether invertebrates use such models remains unclear. Here we examine to what extent prey interception by dragonflies (Plathemis lydia), a behaviour analogous to targeted reaching, requires internal models. By simultaneously tracking the position and orientation of a dragonfly's head and body during flight, we provide evidence that interception steering is driven by forward and inverse models of dragonfly body dynamics and by models of prey motion. Predictive rotations of the dragonfly's head continuously track the prey's angular position. The head-body angles established by prey tracking appear to guide systematic rotations of the dragonfly's body to align it with the prey's flight path. Model-driven control thus underlies the bulk of interception steering manoeuvres, while vision is used for reactions to unexpected prey movements. These findings illuminate the computational sophistication with which insects construct behaviour.