Zero-shot Relation Triplet Extraction (ZSRTE) aims to extract triplets from the context where the relation patterns are unseen during training. Due to the inherent challenges of the ZSRTE task, existing extractive ZSRTE methods often decompose it into named entity recognition and relation classification, which overlooks the interdependence of two tasks and may introduce error propagation. Motivated by the intuition that crucial entity attributes might be implicit in the relation labels, we propose a Relation-Centric joint ZSRTE method named Re-Cent. This approach uses minimal information, specifically unseen relation labels, to extract triplets in one go through a unified model. We develop two span-based extractors to identify the subjects and objects corresponding to relation labels, forming span-pairs. Additionally, we introduce a relation-based correction mechanism that further refines the triplets by calculating the relevance between span-pairs and relation labels. Experiments demonstrate that Re-Cent achieves state-of-the-art performance with fewer parameters and does not rely on synthetic data or manual labor.