News headline generation is a crucial task in increasing productivity for both the readers and producers of news. This task can easily be aided by automated News headline-generation models. However, the presence of irrelevant headlines in scraped news articles results in sub-optimal performance of generation models. We propose that relevance-based headline classification can greatly aid the task of generating relevant headlines. Relevance-based headline classification involves categorizing news headlines based on their relevance to the corresponding news articles. While this task is well-established in English, it remains under-explored in low-resource languages like Telugu due to a lack of annotated data. To address this gap, we present TeClass, the first-ever human-annotated Telugu news headline classification dataset, containing 78,534 annotations across 26,178 article-headline pairs. We experiment with various baseline models and provide a comprehensive analysis of their results. We further demonstrate the impact of this work by fine-tuning various headline generation models using TeClass dataset. The headlines generated by the models fine-tuned on highly relevant article-headline pairs, showed about a 5 point increment in the ROUGE-L scores. To encourage future research, the annotated dataset as well as the annotation guidelines will be made publicly available.