In this paper, we consider a downlink multiuser network operating with finite blocklength (FBL) codes under statistical quality of service (QoS) constraints. Optimal power allocation algorithms are studied to maximize the normalized sum throughput under QoS constraints, while considering different types of data arrivals, namely, constant-rate, Markov, and Markov-modulated Poisson arrivals. We first determine the FBL throughput formulations and subsequently state optimization problems. We show the convexity of the power allocation problem under certain conditions and propose optimal algorithms (for scenarios with different data arrivals). In addition, the FBL performance of equal power allocation and a sub-optimal power allocation algorithm is discussed. Via numerical analysis, we demonstrate the performance improvements with the optimal power allocation. In addition, we provide interesting insights on the system behavior by characterizing the impact of the error probability, the QoS exponent, the blocklength, the number of users, and the source burstiness on the performance.