A function-as-a-service (FaaS) workflow, composed of multiple function services, can realize a complex business application by orchestrating and controlling the function services. The current FaaS workflow execution systems achieve data transfer among function services mainly based on centralized data storages, resulting in heavy data transmission overhead and affecting application performance significantly. In the cases of high concurrency, frequent data transmission will also cause serious contention for network bandwidth resources, resulting in application performance degradation. To address the above problems, this study analyzes the fine-grained data dependency between function services and proposes a critical path-based FaaS workflow deployment optimization method. In addition, the study designs a dependency-sensitive data access and management mechanism to effectively reduce the data transmission between function services, thereby reducing the data transmission latency and end-to-end execution latency of FaaS workflow applications. The study implements a FaaS workflow system, FineFlow, and conducts experiments based on five real-world FaaS workflow applications. The experimental results show that FineFlow can effectively reduce the data transmission latency (the highest reduction and the average reduction are 74.6% and 63.8%, respectively) compared with the FaaS workflow platform with the centralized data storing-based function interaction mechanism. On average, FineFlow reduces the latency of the end-to-end FaaS workflow executions by 19.6%. In particular, for the FaaS workflow application with fine-grained data dependencies, FineFlow can further reduce its data transmission latency and the end-to-end execution latency by 28.4% and 13.8% respectively compared with the state-of-the-art work. In addition, FineFlow can effectively alleviate the impact of network bandwidth fluctuations on application performance by reducing cross-node data transmission, improving the robustness of application performance influenced by the network bandwidth changes.