Although the Gillespie algorithm is accurate in simulating gas-grain reaction networks, so far its computational cost is so expensive that it cannot be used to simulate chemical reaction networks that include molecular hydrogen accretion or the chemical evolution of protoplanetary disks. We present an accelerated Gillespie algorithm that is based on a quasi-steady-state assumption with the further approximation that the population distribution of transient species depends only on the accretion and desorption processes. The new algorithm is tested against a few reaction networks that are simulated by the regular Gillespie algorithm. We found that the less likely it is that transient species are formed and destroyed on grain surfaces, the more accurate the new method is. We also apply the new method to simulate reaction networks that include molecular hydrogen accretion. The results show that surface chemical reactions involving molecular hydrogen are not important for the production of surface species under standard physical conditions of dense molecular clouds.