Aspect Sentiment Triplet Extraction (ASTE) is a subtask of Aspect-Based Sentiment Analysis (ABSA) that aims to extract aspect terms, corresponding opinion terms, and their associated sentiment polarities from text. Current end-to-end approaches, whether employing Large Language Models (LLMs) or complex neural network structures, struggle to effectively model the intricate latent relationships between aspects and opinions. Therefore, in this work, we propose Polarity-Aware Sentiment Triplet Extraction with LLM-as-a-judge (PASTEL), a novel pipeline that decomposes the ASTE task into structured subtasks. We employ finetuned LLMs to separately extract the aspect and opinion terms, incorporating a polarity-aware mechanism to enhance opinion extraction. After generating a candidate set through the Cartesian product of the extracted aspect and opinion-sentiment sets, we leverage an LLM-as-a-Judge to validate and prune these candidates. Experimental evaluations demonstrate that PASTEL outperforms existing baselines. Our findings highlight the necessity of modular decomposition in complex sentiment analysis tasks to fully exploit the capabilities of current LLMs.