We present a simulated cosmology analysis using the second and third moments of the weak lensing mass (convergence) maps. The second moment, or variances, of the convergence as a function of smoothing scale contains information similar to standard shear two-point statistics. The third moment, or the skewness, contains additional non-Gaussian information. The analysis is geared towards the third year (Y3) data from the Dark Energy Survey (DES), but the methodology can be applied to other weak lensing data sets. We present the formalism for obtaining the convergence maps from the measured shear and for obtaining the second and third moments of these maps given partial sky coverage. We estimate the covariance matrix from a large suite of numerical simulations. We test our pipeline through a simulated likelihood analyses varying 5 cosmological parameters and 10 nuisance parameters and identify the scales where systematic or modelling uncertainties are not expected to affect the cosmological analysis. Our simulated likelihood analysis shows that the combination of second and third moments provides a 1.5 per cent constraint on S-8 = sigma(8)(Omega(m)/0.3)(0.5) for DES Year 3 data. This is 20 per cent better than an analysis using a simulated DESY3 shear two-point statistics, owing to the non-Gaussian information captured by the inclusion of higher order statistics. This paper validates our methodology for constraining cosmology with DESYear 3 data, whichwill be presented in a subsequent paper.