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In this paper, we propose a cognitive p-persistent carrier sense multiple access (CpCSMA) scheme for spectrum sharing based cognitive radio (CR) networks. In order to guarantee the quality of service of the primary (licensed) network, secondary users are allowed to transmit their data as long as the interference received at a primary receiver is limited by the predetermined level. In the proposed CpCSMA scheme, secondary users adaptively control the transmit power with ON/OFF fashion according to whether the interference constraint at primary receivers is violated or not. We show that the proposed CpCSMA scheme can be mathematically analyzed by adopting the scaling factor to the conventional p-persistent CSMA analysis framework. The numerical examples illustrate that the analytical results match well to the simulation results and the throughput of the proposed CpCSMA scheme approaches to that of the conventional p- persistent CSMA scheme for high input load. Moreover, the proposed CpCSMA scheme is backward compatible and, thus, can easily be implemented with little modification of the conventional CSMA scheme.