Software debugging is a time-consuming endeavor involving a series of steps, such as fault localization and patch generation, each requiring thorough analysis and a deep understanding of the underlying logic. While large language models (LLMs) demonstrate promising potential in coding tasks, their performance in debugging remains limited. Current LLM-based methods often focus on isolated steps and struggle with complex bugs. In this paper, we propose the first end-to-end framework, FixAgent, for unified debugging through multi-agent synergy. It mimics the entire cognitive processes of developers, with each agent specialized as a particular component of this process rather than mirroring the actions of an independent expert as in previous multi-agent systems. Agents are coordinated through a three-level design, following a cognitive model of debugging, allowing adaptive handling of bugs with varying complexities. Experiments on extensive benchmarks demonstrate that FixAgent significantly outperforms state-of-the-art repair methods, fixing 1.25$\times$ to 2.56$\times$ bugs on the repo-level benchmark, Defects4J. This performance is achieved without requiring ground-truth root-cause code statements, unlike the baselines. Our source code is available on https://github.com/AcceptePapier/UniDebugger.