Rapid nucleic acid testing is central to infectious disease surveillance. Here, we report an assay for rapid COVID-19 testing and its implementation in a prototype microfluidic device. The assay, which we named DISCoVER (for diagnostics with coronavirus enzymatic reporting), involves extraction-free sample lysis via shelf-stable and low-cost reagents, multiplexed isothermal RNA amplification followed by T7 transcription, and Cas13-mediated cleavage of a quenched fluorophore. The device consists of a single-use gravity-driven microfluidic cartridge inserted into a compact instrument for automated running of the assay and readout of fluorescence within 60 min. DISCoVER can detect severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in saliva with a sensitivity of 40 copies μl-1, and was 94% sensitive and 100% specific when validated (against quantitative PCR) using total RNA extracted from 63 nasal-swab samples (33 SARS-CoV-2-positive, with cycle-threshold values of 13-35). The device correctly identified all tested clinical saliva samples (10 SARS-CoV-2-positive out of 13, with cycle-threshold values of 23-31). Rapid point-of-care nucleic acid testing may broaden the use of molecular diagnostics.