With the advancement of information retrieval, recommendation systems, and Retrieval-Augmented Generation (RAG), Approximate Nearest Neighbor Search (ANNS) gains widespread applications due to its higher performance and accuracy. While several disk-based ANNS systems have emerged to handle exponentially growing vector datasets, they suffer from suboptimal performance due to two inherent limitations: 1) failing to overlap SSD accesses with distance computation processes and 2) extended I/O latency caused by suboptimal I/O Stack. To address these challenges, we present FlashANNS, a GPU-accelerated out-of-core graph-based ANNS system through I/O-compute overlapping. Our core insight lies in the synchronized orchestration of I/O and computation through three key innovations: 1) Dependency-Relaxed asynchronous pipeline: FlashANNS decouples I/O-computation dependencies to fully overlap between GPU distance calculations and SSD data transfers. 2) Warp-Level concurrent SSD access: FlashANNS implements a lock-free I/O stack with warp-level concurrency control, to reduce the latency-induced time overhead. 3) Computation-I/O balanced graph degree Selection: FlashANNS selects graph degrees via lightweight compute-to-I/O ratio sampling, ensuring optimal balance between computational load and storage access latency across different I/O bandwidth configurations. We implement FlashANNS and compare it with state-of-the-art out-of-core ANNS systems (SPANN, DiskANN) and a GPU-accelerated out-of-core ANNS system (FusionANNS). Experimental results demonstrate that at $\geq$95\% recall@10 accuracy, our method achieves 2.3-5.9$\times$ higher throughput compared to existing SOTA methods with a single SSD, and further attains 2.7-12.2$\times$ throughput improvement in multi-SSD configurations.