Electronic Raman experiments have shown the presence of two types of gaps in hole-doped cuprate superconductors: one is the gap that increases with underdoping and survives in the pseudogap normal state and the other is the gap that traces the superconducting dome and disappears above the transition temperature. This two-gap behavior is important in that it is related to the mechanism of the pseudogap. By calculating the electronic Raman spectra we show that this behavior is consistent with the picture in which the d-wave superconducting (SC) order and d-density-wave (DDW) order compete in the phase diagram. In particular, the energy of the B1g peak is determined by both the SC and the DDW orders, increases with underdoping and survives in the DDW normal state. On the other hand, the B2g peak is shown to be sensitive to the SC order alone, and thus vanishes in the normal state (even if in the presence of the DDW order). The doping dependence and the temperature dependence of the peak energies in the two channels accord nicely with recent experimental results, which strongly supports the competing-order point of view for the superconducting and pseudogap phases.