Sierra and Summit supercomputers exhibit a significant amount of intranode parallelism between the host POWER9 CPUs and their attached GPU devices. In this article, we show that exploiting device-level parallelism is key to achieving high performance by reducing overheads typically associated with CPU and GPU task execution. Moreover, manually exploiting this type of parallelism in large-scale applications is nontrivial and error-prone. We hide the complexity of exploiting this hybrid intranode parallelism using the OpenMP programming model abstraction. The implementation leverages the semantics of OpenMP tasks to express asynchronous task computations and their associated dependences. Launching tasks on the CPU threads requires a careful design of work-stealing algorithms to provide efficient load balancing among CPU threads. We propose a novel algorithm that removes locks from all task queueing operations that are on the critical path. Tasks assigned to GPU devices require additional steps such as copying input data to GPU devices, launching the computation kernels, and copying data back to the host CPU memory. We perform key optimizations to reduce the cost of these additional steps by tightly integrating data transfers and GPU computations into streams of asynchronous GPU operations. We further map high-level dependences between GPU tasks to the same asynchronous GPU streams to further avoid unnecessary synchronization. Results validate our approach.