An analytical expression for the SAW sensor gas response versus gas concentration (calibration curve) is deduced, at low gas concentrations, on using the perturbation approach. The calibration curve, the sensitivity (its slope) and the detection limit (threshold value) for a given gas and sorbent membrane, can be controlled by a proper choice of the substrate material and its crystallographic orientation. We show how this property takes place from the anisotropy of the SAW propagation on piezoelectric single crystals, allowing to change the partial contributions to the SAW response through mechanical displacements, coupling constant and temperature coefficient of velocity (TCV). The theoretical predictions are experimentally verified on devices implemented on different orientations of quartz and LiNbO3 substrates, using Pd, Pd:Ni and polyvinyl alcohol (PVA) as sorbent films and humid air as analyte. The lowest threshold value of RH (0.01%) and the highest sensitivity (49 ppm/%) are detected in the PVA/yz-LiNbO3 structure, whose sensitivity is about two orders of magnitude higher than that observed in uncoated LiNbO3.