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We investigate electronic transport in graphene nanojunctions and find that the transmission (or the conductance) can exhibit a nonmonotonic behavior with respect to variation in the strength of disorder, mimicking a stochastic resonance. The general setting for this remarkable phenomenon is where the graphene device possesses localized states in the absence of disorder, ie, the localized edge states specific to graphene. A small amount of disorder can then break the localization and lead to an enhancement in the transmission. For strong disorder, Anderson localization [Phys. Rev. 109, 1492 (1958)] sets in, causing the transmission to decrease. The phenomenon is robust and can occur with or without magnetic field.