The mechanism of the trimethylamine or trimethylphosphine catalyzed aza-Morita-Baylis-Hillman (MBH) reaction between acrolein and mesyl imine is investigated by using ab initio and density functional methods. All key transition states are located at the CBS-4M as well as at the mPW1K/6-31+G** levels of theories. To account for the experimentally known rate enhancements through the use of polar protic cocatalysts, transition state models with explicit cocatalysts are considered. Inclusion of polar protic cocatalysts is found to have a profound influence in decreasing the activation barriers associated with the key elementary steps. The protic cocatalysts such as water, methanol, and formic acid are identified as effective in promoting a relay proton transfer. Interestingly, the efficiency of the relay mechanism results in relatively better stabilization of the proton transfer transition state as compared to the addition of enolate to the electrophile (C-C bond formation). The cocatalyst bound models suggest that the proton transfer could become the rate-determining step in the aza-MBH reaction under polar protic conditions. A comparison of the aza-MBH reaction with the analogous MBH reaction is also attempted to bring out the subtle differences between these two reactions. Enhanced kinetic advantages arising from the nature of the activated electrophile are noticed for the aza-MBH reaction. The difference in the relative energies between the transition states for the proton transfer and the C-C bond formation steps with bound cocatalyst(s) is found to be more pronounced in the aza-MBH reaction. In general, the reported results underscore the importance of considering explicit solvents/cocatalysts in order to account for the likely role of the specific interactions between reactants and solvents/cocatalysts.