Cognitive radio inspired non-orthogonal multiple access (CR-NOMA) networks are a research focus in the wireless communication field. The secure communication of wireless networks has become a pressing issue due to the openness of the wireless channel. Multiple-antenna technology can enhance the secrecy performance in CR-NOMA networks, thus we propose a multiple-input-single-output (MISO) CR-NOMA network where the base station (BS) is equipped with multiple antennas, while others have a single antenna in the paper. The BS serves three secondary users in the presence of an eavesdropper and a primary user. We propose two transmit antenna selection (TAS) schemes, namely the space-time transmission (ST) scheme and the maximum channel capacity (MCC) scheme, respectively. Firstly, we obtain an exact closed-form expression for the secrecy outage probability (SOP) of three secondary users and the overall SOP of the networks with the two schemes, respectively. To gain further insights, the present study analyzes the asymptotic SOP performance to analyze the relationship between the network parameters and secrecy outage performance. Based on this, we propose a power allocation algorithm to further improve the secrecy outage performance of the networks. Finally, we verify the analyses with Monte Carlo simulations. The numerical and simulation results demonstrate that: (1) The MCC scheme outperforms the ST scheme on the secrecy outage performance. (2) The proposed power allocation algorithm optimizes the secrecy outage performance of the networks. (3) There exists a sole respective optimal power allocation factor for a BS different transmission power.