The solution of time-dependent PDE-constrained optimization problems subject to unsteady flow equations presents a challenge to both algorithms and computers. In this paper we present an all-at-once approach where we solve for all time-steps of the discretized unsteady Stokes problem at once. The most desirable feature of this approach is that for all steps of an iterative scheme we only need approximate solutions of the discretized Stokes operator. This leads to an efficient scheme which exhibits mesh-independent behaviour.