Recent advances in large-scale generative language models have shown that reasoning capabilities can significantly improve model performance across a variety of tasks. However, the impact of reasoning on a model's ability to mitigate stereotypical responses remains largely underexplored. In this work, we investigate the crucial relationship between a model's reasoning ability and fairness, and ask whether improved reasoning capabilities can mitigate harmful stereotypical responses, especially those arising due to shallow or flawed reasoning. We conduct a comprehensive evaluation of multiple open-source LLMs, and find that larger models with stronger reasoning abilities exhibit substantially lower stereotypical bias on existing fairness benchmarks. Building on this insight, we introduce ReGiFT -- Reasoning Guided Fine-Tuning, a novel approach that extracts structured reasoning traces from advanced reasoning models and infuses them into models that lack such capabilities. We use only general-purpose reasoning and do not require any fairness-specific supervision for bias mitigation. Notably, we see that models fine-tuned using ReGiFT not only improve fairness relative to their non-reasoning counterparts but also outperform advanced reasoning models on fairness benchmarks. We also analyze how variations in the correctness of the reasoning traces and their length influence model fairness and their overall performance. Our findings highlight that enhancing reasoning capabilities is an effective, fairness-agnostic strategy for mitigating stereotypical bias caused by reasoning flaws.