Audio-Visual Speech Recognition (AVSR) integrates acoustic and visual information to enhance robustness in adverse acoustic conditions. Recent advances in Large Language Models (LLMs) have yielded competitive automatic speech recognition performance and shown effectiveness for AVSR. However, prior approaches project audio and visual features independently or apply shallow fusion, limiting cross-modal alignment and complementary exchange while increasing the LLM's computational load. To address this, we propose AVUR-LLM, an LLM-based Audio-Visual Speech Recognition via Sparse Modality Alignment and Visual Unit-Guided Refinement. Experiments on LRS3 demonstrate state-of-the-art results for AVSR. Under additive-noise conditions at 0 dB SNR, it achieves 37% relative improvement over the baseline system.