Fine-tuning language models is commonly believed to inevitably harm their safety, i.e., refusing to respond to harmful user requests, even when using harmless datasets, thus requiring additional safety measures. We challenge this belief through systematic testing, showing that poor optimization choices, rather than inherent trade-offs, often cause safety problems, measured as harmful responses to adversarial prompts. By properly selecting key training hyper-parameters, e.g., learning rate, batch size, and gradient steps, we reduce unsafe model responses from 16\% to approximately 5\%, as measured by keyword matching, while maintaining utility performance. Based on this observation, we propose a simple exponential moving average (EMA) momentum technique in parameter space that preserves safety performance by creating a stable optimization path and retains the original pre-trained model's safety properties. Our experiments on the Llama families across multiple datasets (Dolly, Alpaca, ORCA) demonstrate that safety problems during fine-tuning can largely be avoided without specialized interventions, outperforming existing approaches that require additional safety data while offering practical guidelines for maintaining both model performance and safety during adaptation.