Recent years have witnessed video streaming gradually evolve into one of the most popular Internet applications. With the rapidly growing personalized demand for realtime video streaming services, maximizing their Quality of Experience (QoE) is a long-standing challenge. The emergence of the serverless computing paradigm has potential to meet this challenge through its fine-grained management and highly parallel computing structures. However, it is still ambiguous how to implement and configure serverless components to optimize video streaming services. In this paper, we propose EAVS, an Edge-assisted Adaptive Video streaming system with Serverless pipelines, which facilitates fine-grained management for multiple concurrent video transmission pipelines. Then, we design a chunk-level optimization scheme to solve the video bitrate adaptation issue. We propose a Deep Reinforcement Learning (DRL) algorithm based on Proximal Policy Optimization (PPO) with a trinal-clip mechanism to make bitrate decisions efficiently for better user-perceived QoE. Finally, we implement the serverless video streaming system prototype and evaluate the performance of EAVS on various real-world network traces. Extensive results show that our proposed EAVS significantly improves userperceived QoE and reduces the video stall rate, achieving over 9.1% QoE improvement and 60.2% latency reduction compared to state-of-the-art solutions.