We present an adaptive control scheme for the control of Markov chains to minimize long-run average cost when the system transition and reward structures are unknown. Q-factors are estimated along a single sample path of the system and control actions are applied based on the latest estimates, We prove that an optimal policy is obtained asymptotically with probability one. More importantly, ne prove that optimal system performance is achieved as well, which means that the performance of the system can not be bettered even if the system transition and reward structures are known. An example is given to illustrate our adaptive control scheme.