There is an increasing demand for robust, miniaturized sensors with ppm or parts per 109 (ppb) sensing capability, and high selectivity to different chemical or biological species. Here we show that trace amounts (ppb) of gases or organic solvent vapors can be detected with high selectivity and sensitivity using single-walled carbon nanotube bundles in a resonator configuration. The enhanced sensing properties result from a change in the effective dielectric properties of the resonator when exposed to different gas environments. A theoretical model is described which computes resonant frequency shifts that are in remarkable agreement with corresponding experimental shifts exhibited by the resonator when exposed to different gas molecules. This work demonstrates a gas-sensing platform with superior sensitivity and selectivity for gas detection, and presents advantages in terms of portability and recovery time. In particular, the sensing platform does not require functionalized carbon nanotubes to enhance specificity, or wire connection to the nanotubes making it attractive for remote sensor technology.