Large language model (LLM)-based applications consist of both LLM and non-LLM components, each contributing to the end-to-end latency. Despite great efforts to optimize LLM inference, end-to-end workflow optimization has been overlooked. Existing frameworks employ coarse-grained orchestration with task modules, which confines optimizations to within each module and yields suboptimal scheduling decisions. We propose fine-grained end-to-end orchestration, which utilizes task primitives as the basic units and represents each query's workflow as a primitive-level dataflow graph. This explicitly exposes a much larger design space, enables optimizations in parallelization and pipelining across primitives of different modules, and enhances scheduling to improve application-level performance. We build Teola, a novel orchestration framework for LLM-based applications that implements this scheme. Comprehensive experiments show that Teola can achieve up to 2.09x speedup over existing systems across various popular LLM applications.