In this paper, ethylene-vinyl acetate-glycidyl methacrylate terpolymer elastomer (EVM-GMA) was vulcanized by methyltetrahydrophthalic anhydride (MTHPA) and cross-linked by dicumyl peroxide (DCP) as a reference. Mechanical properties and hot-air aging properties were investigated. The cross-linking mechanism of EVM-GMA reacting with MTHPA was proposed, and a "model vulcanization reaction' was designed to prove its rationality. The reaction products were characterized by GPC, FT-IR, GC-MS, H-1-NMR and so on. The results indicated that the cured torque M-H - M-L and cross-link density of EVM-GMA increased first and then decreased with the increase in amount of MTHPA. An optimum of curing, mechanical properties and aging resistance were found at a molar ratio of anhydride to GMA of 1:2. The MTHPA curing reaction catalyzed by tetrabutylammonium bromide was faster compared to the selected DCP cure system, and the mechanical properties, compression set and hot-air aging performance were superior as well. The "model vulcanization reaction" indicated that one molecule of MTHPA successfully reacted with two molecules of the mono-epoxy compound of glycidyl-versatic ester (E10P) to form a hydroxyl-terminated epoxy-anhydride-epoxy polyester structure. Therefore, one molecule of MTHPA can "bridge" two epoxy side groups (provided by GMA) on the adjacent molecular chain of EVM-GMA to realize rubber cross-linking, which proved that the cross-linking mechanism based on epoxy-anhydride reaction for EVM-GMA elastomer proposed in this paper was reasonable.