In this paper, the problem of efficient distributed power control via convex pricing of users' transmission power in the uplink of CDMA wireless networks supporting multiple services is addressed. Each user is associated with a nested utility function, which appropriately represents his degree of satisfaction in relation to the expected trade-off between his QoS-aware actual uplink throughput performance and the corresponding power consumption. Initially, a Multiservice Uplink Power Control game (MSUPC) is formulated, where each user aims selfishly at maximizing his utility-based performance under the imposed physical limitations and its unique Nash equilibrium point is determined. Then the inefficiency of MSUPC game's Nash equilibrium is proven and a usage-based convex pricing policy of the transmission power is introduced, which offers a more effective approach compared to the linear pricing schemes that have been adopted in the literature. Consequently, a Multiservice Uplink Power Control game with Convex Pricing (MSUPC-CP) is formulated and its unique Pareto optimal Nash equilibrium is determined. A distributed iterative algorithm for computing MSUPC-CP game's equilibrium is proposed, while the overall approach's efficiency is illustrated via modeling and simulation.