The excitation of a quantum oscillator at transitions between stationary states by wavelet pulses with zero and nonzero area as a function of their duration and amplitude is studied. The excitation from the ground state is examined in detail, together with the specific features of this process during excitation from excited states. Analytical expressions for the main characteristics of excitation probability under weak and strong perturbation of quantum oscillator were obtained within limits of strict consideration. Particularly, it was found that, on a qualitative and quantitative level, there is no difference between excitation by pulses with zero and non-zero area.