The presence of massive neutrinos affects structure formation, leaving imprints on large-scale structure observables such as the weak lensing field. The common lensing analyses with two-point statistics are insensitive to the large amount of non-Gaussian information in the density field. We investigate non-Gaussian tools, in particular the Minkowski Functionals (MFs)—morphological descriptors including area, perimeter, and genus—in an attempt to recover the higher-order information. We use convergence maps from the Cosmological Massive Neutrino Simulations (MassiveNus) and assume galaxy noise, density, and redshift distribution for an LSST-like survey. We show that MFs are sensitive to the neutrino mass sum through non-Gaussian features of the lensing field, with a redshift dependence different from that of the power spectrum. We find that redshift tomography significantly improves the constraints on neutrino mass for MFs, compared to the improvements for the convergence power spectrum. We attribute this to the stronger redshift dependence of neutrino effects on small scales. We then build an emulator to model the power spectrum and MFs, and study the constraints on [Mν, Ωm, As] from the power spectrum, MFs, and their combination. We show that, for weak lensing convergence, MFs significantly outperform the power spectrum in constraining neutrino mass, by more than a factor of four. However, a thorough study of the impact from systematics such as baryon physics, galaxy shape and redshift biases will be important to realize the full potential of MFs.