Although N-body simulations of cosmic structure formation suggest that dark matter halos have density profiles shallower than isothermal at small radii and steeper at large radii, whether observed galaxy clusters follow this profile is still ambiguous. We use one such density profile, the asymmetric Navarro-Frenk-White (NFW) profile, to model the mass distributions of 10 galaxy clusters with gravitational arcs observed by the Hubble Space Telescope (HST). We characterize the galaxy lenses in each cluster as NFW ellipsoids, each defined by an unknown scale convergence, scale radius, ellipticity, and position angle. For a given set of values of these parameters, we compute the arcs that would be produced by such a lens system. To define the goodness of fit to the observed arc system, we define a χ2 function encompassing the overlap between the observed and reproduced arcs, as well as the agreement between the predicted arc sources and the observational constraints on the source system. We minimize this χ2 to find the values of the lens parameters that best reproduce the observed arc system in a given cluster. Here we report our best-fit lens parameters and corresponding mass estimates for each of the 10 lensing clusters. We find that cluster mass models based on lensing galaxies defined as NFW ellipsoids can accurately reproduce the observed arcs and that the best-fit parameters to such a model fall within the reasonable ranges defined by simulations. These results assert NFW profiles as an effective model for the mass distributions of observed clusters.