Pristine graphene and h-BN monolayers are chemically inert to oxygen and thus exhibit very limited catalytic activity toward O2 activation. Herein, we show that graphene/h-BN heterostructures exhibit a surprising O2 activation capability. We theoretically designed ten graphene/h-BN heterostructures with three types of interfaces and investigated their catalytic activities toward O2 activation and CO-oxidation. In general, O2 can be molecularly chemisorbed and activated on electron-rich graphene/h-BN heterostructures. Electron-deficient graphene/h-BN heterostructures can lead to dissociative O2 adsorption with relatively low dissociation energy barriers (<0.4 eV). For CO-oxidation, the computed energy barrier can be as low as 0.67 eV. The high catalytic activities toward O2 stem from either electron-deficient heterostructures' accumulated electrons or electron richness and low work function for the electron-rich heterostructures. Although the catalytic activities of graphene/h-BN heterostructures depend strongly on the interface type, they are insensitive to the patterns of BN-substitutes, hence benefiting applicability of a wide range of heterostructures.