Using a compactness-uniqueness approach, we show that the Fattorini criterion implies the exact controllability of general compactly perturbed controlled linear systems. We then apply this perturbation result to obtain new controllability results for systems governed by partial differential equations. Notably, we combine it with the fictitious control method to establish the exact controllability of a cascade system of coupled wave equations by a reduced number of controls and with the same control time as the one for a single wave equation. We also combine this perturbation result with transmutation techniques to prove the null controllability in arbitrarily small time of a one-dimensional non diagonalizable system of coupled heat equations with as many controls as equations.