Fine-tuning is an essential and pervasive functionality for applying large language models (LLMs) to downstream tasks. However, it has the potential to substantially degrade safety alignment, e.g., by greatly increasing susceptibility to jailbreak attacks, even when the fine-tuning data is entirely harmless. Despite garnering growing attention in defense efforts during the fine-tuning stage, existing methods struggle with a persistent safety-utility dilemma: emphasizing safety compromises task performance, whereas prioritizing utility typically requires deep fine-tuning that inevitably leads to steep safety declination. In this work, we address this dilemma by shedding new light on the geometric interaction between safety- and utility-oriented gradients in safety-aligned LLMs. Through systematic empirical analysis, we uncover three key insights: (I) safety gradients lie in a low-rank subspace, while utility gradients span a broader high-dimensional space; (II) these subspaces are often negatively correlated, causing directional conflicts during fine-tuning; and (III) the dominant safety direction can be efficiently estimated from a single sample. Building upon these novel insights, we propose safety-preserving fine-tuning (SPF), a lightweight approach that explicitly removes gradient components conflicting with the low-rank safety subspace. Theoretically, we show that SPF guarantees utility convergence while bounding safety drift. Empirically, SPF consistently maintains downstream task performance and recovers nearly all pre-trained safety alignment, even under adversarial fine-tuning scenarios. Furthermore, SPF exhibits robust resistance to both deep fine-tuning and dynamic jailbreak attacks. Together, our findings provide new mechanistic understanding and practical guidance toward always-aligned LLM fine-tuning.