Task-oriented dialogue (TOD) systems facilitate goal-driven interactions between users and machines. While recent advances in deep learning have improved the performance, TOD systems often struggle in low-resource scenarios with limited labeled data. To address this challenge, we propose Spec-TOD, a novel framework designed to train an end-to-end TOD system with limited data. Spec-TOD introduces two main innovations: (i) a novel specialized end-to-end TOD framework that incorporates explicit task instructions for instruction-tuned large language models (LLMs), and (ii) an efficient training strategy that leverages lightweight, specialized LLMs to achieve strong performance with minimal supervision. Experiments on the MultiWOZ dataset, a widely used TOD benchmark, demonstrate that Spec-TOD achieves competitive results while significantly reducing the need for labeled data. These findings highlight the potential of the proposed framework in advancing efficient and effective TOD systems in low-resource settings.