As an introduction of a kinematic survey of Magellanic Cloud (MC) star clusters, we report on the dynamical masses and mass-to-light ratios (M/L) of NGC 419 (Small Magellanic Cloud) and NGC 1846 (Large Magellanic Cloud). We have obtained more than one hundred high-resolution stellar spectra in and around each cluster using the multi-object spectrograph M2FS on the Magellan/Clay Telescope. Line-of-sight velocities and positions of the stars observed in each cluster were used as input to an expectation-maximization algorithm used to estimate cluster membership probabilities, resulting in samples of 46 and 52 likely members (P_M_>=50 per cent) in NGC 419 and NGC 1846, respectively. This process employed single-mass King models constrained by the structural parameters of the clusters and provided self-consistent dynamical mass estimates for both clusters. Our best-fitting results show that NGC 419 has a projected central velocity dispersion of 2.44^+0.37^_-0.21_km/s, corresponding to a total mass of 7.6^+2.5^_-1.3_x10^4^M_{sun}_ and V-band M/L ratio of 0.22^+0.08^_-0.05_ in solar units. For NGC 1846, the corresponding results are 2.04^+0.28^_-0.24_km/s, 5.4^+1.5^_-1.4_x10^4^M_{sun}_, and 0.32^+0.11^_-0.11_. The mean metallicities of NGC 419 and NGC 1846 are found to be [Fe/H]=-0.84+/-0.19 and -0.70+/-0.08, respectively, based on the spectra of likely cluster members. We find marginal statistical evidence of rotation in both clusters, though in neither cluster does rotation alter our mass estimates significantly. We critically compare our findings with those of previous kinematic studies of these two clusters in order to evaluate the consistency of our observational results and analytic tools.