We consider an optimal control problem for a dynamical system described by a Caputo fractional differential equation of order α∈(0,1) and a terminal cost functional. We prove that, under certain assumptions, the (non-smooth, in general) value functional of this problem has a property of directional differentiability of order α. As an application of this result, we propose a new method for constructing an optimal positional (feedback) control strategy, which allows us to generate ε-optimal controls for any predetermined accuracy ε>0.