We present a pilot, untargeted extragalactic carbon monoxide (CO) emission-line survey using ALMACAL, a project utilizing ALMA calibration data for scientific purposes. In 33 deep (T-exp' > 40 min) ALMACAL fields, we report six CO emission-line detections above S/N > 4, one-third confirmed by MUSE observations. With this pilot survey, we probe a cosmologically significant volume of similar to 10(5) cMpc(3), widely distributed over many pointings in the southern sky, making the survey largely insusceptible to the effects of cosmic variance. We derive the redshift probability of the CO detections using probability functions from the SHARK semi-analytical model of galaxy formation. By assuming typical CO excitations for the detections, we put constraints on the cosmic molecular gas mass density evolution over the redshift range 0 < z < 1.5. The results of our pilot survey are consistent with the findings of other untargeted emission-line surveys and the theoretical model predictions and currently cannot rule out a non-evolving molecular gas mass density. Our study demonstrates the potential of using ALMA calibrator fields as a multi-sightline untargeted CO emission-line survey. Applying this approach to the full ALMACAL database will provide an accurate, free of cosmic variance, measurement of the molecular luminosity function as a function of redshift.