We investigate the effect of slowly-varying parameter on the energy transfer in a system of weakly coupled nonlinear oscillators, with special attention to a mathematical analogy between the classical energy transfer and quantum transitions. For definiteness, we consider a system of two weakly coupled oscillators with cubic nonlinearity, in which the oscillator with constant parameters is excited by an initial impulse, while a coupled oscillator with slowly-varying parameters is initially at rest. It is proved that the equations of the slow passage through resonance in this system are identical to equations of the nonlinear Landau-Zener (LZ) tunneling. Three types of dynamical behavior are distinguished, namely, quasi-linear, moderately nonlinear and strongly nonlinear. The quasi-linear systems exhibit a gradual energy transfer from the excited to the attached oscillator, while the moderately nonlinear systems are characterized by an abrupt transition from the energy localization on the excited oscillator to the localization on the attached oscillator. In the strongly nonlinear systems, the transition from the energy localization to strong energy exchange between the oscillators is revealed. A special case of the rapid irreversible energy transfer in the strongly nonlinear system with slowly-varying parameters is also investigated. The conditions providing different types of the dynamical behavior are derived. Explicit approximate solutions describing the transient processes in moderately and strongly nonlinear systems are suggested. Correctness of the constructed approximations is confirmed by numerical results.