Challenging applications in trace gas measurements require low uncertainty and high acquisition rates. Many cavity-enhanced spectroscopies exhibit significant sensitivity and potential, but their scanning rates are limited by reliance on either mechanical or thermal frequency tuning. Here, we present frequency-agile, rapid scanning spectroscopy (FARS) in which a high-bandwidth electro-optic modulator steps a selected laser sideband to successive optical cavity modes. This approach involves no mechanical motion and allows for a scanning rate of 8 kHz per cavity mode, a rate that is limited only by the cavity response time itself. Unlike rapidly frequency-swept techniques, FARS does not reduce the measurement duty cycle, degrade the spectrum's frequency axis or require an unusual cavity configuration. FARS allows for a sensitivity of ~2 × 10-12 cm-1 Hz-1/2 and a tuning range exceeding 70 GHz. This technique shows promise for fast and sensitive trace gas measurements and studies of chemical kinetics.