Rumors can spread very rapidly through online social networks (OSNs), leading to huge impact on human society. Pushing anti-rumor messages to OSN users proves to be an effective approach to mitigating the impact of rumors. This paper focuses on the problem of finding a cost-effective anti-rumor message-pushing scheme. First, based on a node-level rumor spreading model with anti-rumor message-pushing mechanism, we estimate the cost effectiveness of an anti-rumor message-pushing scheme and thereby model the original problem as an optimal control problem. Second, we show that the optimal control problem admits an optimal control. Thirdly, we derive the optimality system for the optimal control problem. Finally, we find that a control obtained by solving the optimality system is cost-effective. This work contributes to containing the prevalence of rumors in a cost-effective way.