We investigate theoretically the nonresonant excitation of vibrational levels in polar molecules by unipolar radiation pulses of duration much shorter than the characteristic period of the molecule's vibration. We consider several profiles of the potential of the interaction of atoms in a diatomic molecule and derive analytically the probabilities of the molecule's transition to excited vibrational states when driven by subcycle unipolar pulses. It is shown that the excitation efficiency is governed by the electric pulse area so that unipolar half-cycle pulses turn out to be the most efficient ones. We introduce the characteristic scale of the electric pulse area, which serves as a measure of the pulse action on the vibrational states of the molecule. The results are generalized to the interaction of excited vibrational and rotational states and it is shown that the behavior of the vibrational levels' populations versus the electric pulse area as well as the introduced characteristic scale stays valid.