A one-to-one relation between a variable-coefficient (3+1)-dimensional nonlinear Schrödinger equation with linear and parabolic potentials and the standard nonlinear Schrödinger equation is presented, and then superposed rogue-wave-like breather solution is obtained. These explicit expressions, describing the evolution of the amplitude, width, center and phase, imply that the diffraction, nonlinearity and gain/loss parameters interplay together to influence evolutional characteristics above. Moreover, the controllable mechanism for fast excitation, maintenance, restraint and recurrence of breather is studied. We also provide an experimental scheme to observe these phenomena in future experiments.