This work introduces non-orthogonal multiple access (NOMA) into short-packet communications to reduce latency over the air interface in wireless networks. Specifically, we address the optimization of transmission rates and power allocation to maximize the effective throughput of the user with a higher channel gain while guaranteeing the other user achieving a certain level of effective throughput. To demonstrate the benefits of NOMA, we analyze the optimal performance of orthogonal multiple access (OMA) as a benchmark. Our examination shows that NOMA can significantly outperform OMA by achieving a higher effective throughput with the same latency or incurring a lower latency to achieve the same effective throughput targets. Surprisingly, we find that the performance gap between NOMA and OMA becomes more prominent when the effective throughput targets at the two users become comparable. This demonstrates that NOMA can significantly reduce the transmission latency and achieve fairness between users for short-packet communications.