We study the cosmological information contained in the Minkowski Functionals ( MFs ) of weak gravitational lensing convergence maps . We show that the MFs provide strong constraints on the local type primordial non-Gaussianity parameter f _ { { NL } } . We run a set of cosmological N -body simulations and perform ray-tracing simulations of weak lensing , to generate 100 independent convergence maps of 25 deg ^ { 2 } field-of-view for f _ { NL } = -100 , 0 and 100 . We perform a Fisher analysis to study the degeneracy among other cosmological parameters such as the dark energy equation of state parameter w and the fluctuation amplitude \sigma _ { 8 } . We use fully nonlinear covariance matrices evaluated from 1000 ray-tracing simulations . For the upcoming wide-field observations such as Subaru Hyper Suprime-Cam survey with the proposed survey area of 1500 deg ^ { 2 } , the primordial non-Gaussianity can be constrained with a level of f _ { { NL } } \sim 80 and w \sim 0.036 by weak lensing MFs . If simply scaled by the effective survey area , a 20000 deg ^ { 2 } lensing survey using Large Synoptic Survey Telescope will give constraints of f _ { { NL } } \sim 25 and w \sim 0.013 . We show that these constraints can be further improved by a tomographic method using source galaxies in multiple redshift bins .