Galaxy-wide star formation can be quenched by a number of physical processes such as environmental effects (e.g. ram pressure stripping) and supernova feedback. Using numerical simulations, we here demonstrate that star formation can be severely suppressed in disc galaxies with their gas discs counter-rotating with respect to their stellar discs. This new mechanism of star formation suppression (or quenching) does not depend so strongly on model parameters of disc galaxies, such as bulge-to-disc ratios and gas mass fractions. Such severe suppression of star formation is largely due to the suppression of the gas density enhancing mechanism, i.e. spiral arm formation in disc galaxies with counter-rotating gas. Our simulations also show that molecular hydrogen and dust can be rather slowly consumed by star formation in disc galaxies with counter-rotating gas discs (i.e. long gas depletion time-scale). Based on these results, we suggest that spiral and S0 galaxies with counter-rotation can have rather low star formation rate for their gas densities. Also, we suggest that a minor fraction of S0 galaxies have no prominent spiral arms because they have a higher fraction of counter-rotating gas. We predict that poststarburst 'E+A' disc galaxies with cold gas could have counter-rotating gas.