A new random access scheme is proposed to solve the intra-cell pilot collision for M2M communication in crowded asynchronous massive multiple-input multiple-output (MIMO) systems. The proposed scheme utilizes the proposed estimation of signal parameters via rotational invariance technique enhanced (ESPRIT-E) method to estimate the effective timing offsets, and then active UEs obtain their timing errors from the effective timing offsets for uplink message transmission. We analyze the mean squared error of the estimated effective timing offsets of UEs, and the uplink throughput. Simulation results show that, compared to the exiting random access scheme for the crowded asynchronous massive MIMO systems, the proposed scheme can improve the uplink throughput and estimate the effective timing offsets accurately at the same time.