We consider the class of nonlinear, autonomous control systems on a differentiable manifold that depend semilinearly on the control variable. This class includes the much-studied class of control-linear systems. For such control-semilinear systems, we prove that global controllability is preserved under nonlinear perturbations satisfying a mild boundedness condition. Our techniques enable us to obtain controllability results for piecewise-constant and smooth controls, as well as for measurable controls.