Large Language Models (LLMs) have demonstrated exceptional promise in translation tasks for high-resource languages. However, their performance in low-resource languages is limited by the scarcity of both parallel and monolingual corpora, as well as the presence of noise. Consequently, such LLMs suffer with alignment and have lagged behind State-of-The-Art (SoTA) neural machine translation (NMT) models in these settings. This paper introduces NusaMT-7B, an LLM-based machine translation model for low-resource Indonesian languages, starting with Balinese and Minangkabau. Leveraging the pretrained LLaMA2-7B, our approach integrates continued pre-training on monolingual data, Supervised Fine-Tuning (SFT), self-learning, and an LLM-based data cleaner to reduce noise in parallel sentences. In the FLORES-200 multilingual translation benchmark, NusaMT-7B outperforms SoTA models in the spBLEU metric by up to +6.69 spBLEU in translations into Balinese and Minangkabau, but underperforms by up to -3.38 spBLEU in translations into higher-resource languages. Our results show that fine-tuned LLMs can enhance translation quality for low-resource languages, aiding in linguistic preservation and cross-cultural communication.