Recent studies have demonstrated the superiority of non-orthogonal multiple access (NOMA) over orthogonal multiple access (OMA) in cooperative communication networks. In this paper, we propose a novel half-duplex cooperative asynchronous NOMA (C-ANOMA) framework with user relaying, where a timing mismatch is intentionally added in the broadcast signal. We derive the expressions for the individual throughputs of the strong user (acts as relay) which employs the block-wise successive interference cancellation (SIC) and the weak user which combines the symbol-asynchronous signal with the interference-free signal. We analytically prove that in the C-ANOMA systems with a sufficiently large frame length, the strong user attains the same throughput to decode its own message while both users can achieve a higher throughput to decode the weak user's message compared with those in the cooperative NOMA (C-NOMA) systems. Besides, we obtain the optimal timing mismatch when the frame length goes to infinity. Furthermore, to exploit the trade-off between the power consumption of base station and that of the relay user, we solve a weighted sum power minimization problem under quality of services (QoS) constraints. Numerical results show that the C-ANOMA system can consume less power compared with the C-NOMA system to satisfy the same QoS requirements.