The integration of language and vision for object affordance understanding is pivotal for the advancement of embodied agents. Current approaches are often limited by reliance on segregated pre-processing stages for language interpretation and object localization, leading to inefficiencies and error propagation in affordance segmentation. To overcome these limitations, this study introduces a unique task, part-level affordance grounding, in direct response to natural language instructions. We present the Instruction-based Affordance Grounding Network (IAG-Net), a novel architecture that unifies language–vision interactions through a varied-scale multimodal attention mechanism. Unlike existing models, IAG-Net employs two textual–visual feature fusion strategies, capturing both sentence-level and task-specific textual features alongside multiscale visual features for precise and efficient affordance prediction. Our evaluation on two newly constructed vision–language affordance datasets, ITT-AFF VL and UMD VL, demonstrates a significant leap in performance, with an improvement of 11.78% and 0.42% in mean Intersection over Union (mIoU) over cascaded models, bolstering both accuracy and processing speed. We contribute to the research community by releasing our source code and datasets, fostering further innovation and replication of our findings.