Aspartate proteases are potential targets for various diseases, and many of their inhibitors are FDA-approved drugs. However, these peptidomimetic and reversibly bound drugs become ineffective upon prolonged use. Attempts have been made to design and synthesize various nonpeptidic epoxide-based irreversible inhibitors to combat the drug-resistance enigma. Here, we study the mechanism of epoxide ring opening in two widely studied aspartate proteases, HIV-1 protease and pepsin. Our results from QM/MM molecular dynamics show that the epoxide ring opening in aspartate proteases follow a two-step mechanism with the formation of an oxyanion intermediate, stabilized by a set of water molecules in the protein active site. These water molecules by virtue of "low-barrier hydrogen bonds" with the epoxide ring reduce the intrinsic reaction barrier while remaining structurally unperturbed, thus playing a cocatalytic role. We validated our results by reproducing the experimentally observed protease/pepsin-epoxide covalent complexes as end products. The observed stability of our oxyanion intermediate in a four-water-coordinated state is also consistent with the reported stable state of the hydroxide ion in water as OH-(H2O)4. Our study could pave the way for the design of new class "HIV protease irreversible inhibitors" from the acquired knowledge of the structures of intermediate and transition states traced during the explored reaction mechanism.