A styrene-butadiene-styrene co-polymer matrix nanocomposite filled with graphene nanoplatelets was studied to prepare chemiresistive volatile organic compounds (VOCs) room temperature sensors with considerable response and selectivity. Nanofiller concentration was estimated from the electrical conductivity percolation behaviour of the nanocomposite. Fabricated sensors provided selective relative responses to representative VOCs differing by orders of magnitude. Maximum observed average relative responses upon exposure to saturated vapours of the tested VOCs were ca. 23% for ethanol, 1600% for acetone, and the giant values were 9 × 106% for n-heptane and 10 × 106% for toluene. The insensitivity of the sensor to the direct saturated water vapour exposure was verified. Although high humidity decreases the sensor’s response, it paradoxically enhances the resolution between hydrocarbons and polar organics. The non-trivial sensing mechanism is explained using the Hansen solubility parameters (HSP), enabling a rational design of new sensors; thus, the HSP-based class of sensors is outlined.