Automatically configuring storage systems is hard: parameter spaces are large and conditions vary across workloads, deployments, and versions. Heuristic and ML tuners are often system specific, require manual glue, and degrade under changes. Recent LLM-based approaches help but usually treat tuning as a single-shot, system-specific task, which limits cross-system reuse, constrains exploration, and weakens validation. We present StorageXTuner, an LLM agent-driven auto-tuning framework for heterogeneous storage engines. StorageXTuner separates concerns across four agents - Executor (sandboxed benchmarking), Extractor (performance digest), Searcher (insight-guided configuration exploration), and Reflector (insight generation and management). The design couples an insight-driven tree search with layered memory that promotes empirically validated insights and employs lightweight checkers to guard against unsafe actions. We implement a prototype and evaluate it on RocksDB, LevelDB, CacheLib, and MySQL InnoDB with YCSB, MixGraph, and TPC-H/C. Relative to out-of-the-box settings and to ELMo-Tune, StorageXTuner reaches up to 575% and 111% higher throughput, reduces p99 latency by as much as 88% and 56%, and converges with fewer trials.