We have used recent observational data for the Cepheids in the rich, young LMC clusters NGC 1866 and NGC 2031 to constrain the cluster distances with the mass-equivalency (ME) method. The basis of this approach is to fix the cluster distances by requiring the Cepheid evolutionary and pulsational masses to be equal. Using evolutionary models incorporating a mild amount of core and envelope overshooting along with recent pulsational models, we derive distance moduli of 18.51 +/- 0.21 and 18.32 +/- 0.20 for NGC 1866 and NGC 2031, respectively. The quoted errors are dominated by the uncertainties in the heavy element abundances of the clusters (assumed to be 0.3 dex for both clusters), with a smaller contribution due to the apparently intrinsic spread in the masses of the Cepheids in each cluster. For the ME method, we find that Delta(m-M)(0)/Delta sub Z(1) = 0.69, where Z(1) = log (Z/0.016). This result implies that the cluster distances can be determined to better than +/- 5 percent if the cluster abundances can be measured to better than about +/- 0.15 dex.