The translation of idiomatic expressions often results in misunderstandings and inaccuracies, affecting everyday communication as well as machine translation systems. This paper introduces Idiom-aware Vietnamese Translation (IDiAT), a new framework for the evaluation of idiomatic translation for Vietnamese, along with state-of-the-art results for this task. We collect and curate a high-quality Vietnamese-English idiom set that serves as a resource for in-context learning (ICL). IDiAT's evaluation benchmark includes both idiomatic and non-idiomatic text pairs to assess general translation quality and idiomatic translation performance. We leverage ICL in large language models to augment few-shot demonstrations with idiom and topic descriptions and consequently improve the translation accuracy. Empirical results demonstrate that our IDiAT-based ICL outperforms traditional supervised methods using only a few data samples. Multiple evaluations confirm the effectiveness of our proposed approach. Though focusing on the Vietnamese language, our approach advances idiomatic translation and contributes to the development of culturally aware translation systems, paving the way for future research in low-resource languages. The experimental materials are publicly available.