Below- and near-threshold harmonic generation provides a potential approach to achieve a high conversion efficiency of vacuum-ultraviolet and extreme-ultraviolet sources for the advancement of spectroscopy. Here, we perform a time-dependent density functional theory study for the nonperturbative treatment of below- and near-threshold harmonic generation of CO and N2 diatomic molecules subject to short near-infrared laser pulses and aligned parallel to the laser field polarization. We find that with the use of different driving laser pulse shapes, we can control and enhance harmonic generation through the excited-state resonance structures. Depending on the pulse shape, the enhancement can reach five to seven orders of magnitude as compared to the reference sine-squared laser pulse of the same duration. The results for different driving laser intensities are also presented and discussed in detail.