We show that vapors of different chemicals produce distinguishably different effects on the low-frequency noise spectra of graphene. It was found in a systematic study that some gases change the electrical resistance of graphene devices without changing their low-frequency noise spectra while other gases modify the noise spectra by inducing Lorentzian components with distinctive features. The characteristic frequency f(c) of the Lorentzian noise bulges in graphene devices is different for different chemicals and varies from f(c) = 10-20 Hz to f(c) = 1300-1600 Hz for tetrahydrofuran and chloroform vapors, respectively. The obtained results indicate that the low-frequency noise in combination with other sensing parameters can allow one to achieve the selective gas sensing with a single pristine graphene transistor. Our method of gas sensing with graphene does not require graphene surface functionalization or fabrication of an array of the devices with each tuned to a certain chemical.