Neural bidirectional reflectance distribution functions (BRDFs) have emerged as popular material representations for enhancing realism in physically-based rendering. Yet their importance sampling remains a significant challenge. In this paper, we introduce a reparameterization-based formulation of neural BRDF importance sampling that seamlessly integrates into the standard rendering pipeline with precise generation of BRDF samples. The reparameterization-based formulation transfers the distribution learning task to a problem of identifying BRDF integral substitutions. In contrast to previous methods that rely on invertible networks and multi-step inference to reconstruct BRDF distributions, our model removes these constraints, which offers greater flexibility and efficiency. Our variance and performance analysis demonstrates that our reparameterization method achieves the best variance reduction in neural BRDF renderings while maintaining high inference speeds compared to existing baselines.