The gene set analysis (GSA) is a foundational approach for uncovering the molecular functions associated with a group of genes. Recently, LLM-powered methods have emerged to annotate gene sets with biological functions together with coherent explanatory insights. However, existing studies primarily focus on proprietary models, which have been shown to outperform their open-source counterparts despite concerns over cost and data privacy. Furthermore, no research has investigated the application of advanced reasoning strategies to the GSA task. To address this gap, we introduce Gene-R1, a data-augmented learning framework that equips lightweight and open-source LLMs with step-by-step reasoning capabilities tailored to GSA. Experiments on 1,508 in-distribution gene sets demonstrate that Gene-R1 achieves substantial performance gains, matching commercial LLMs. On 106 out-of-distribution gene sets, Gene-R1 performs comparably to both commercial and large-scale LLMs, exhibiting robust generalizability across diverse gene sources.