This paper presents a novel multistage fine-tuning strategy designed to enhance automatic speech recognition (ASR) performance in low-resource languages using OpenAI's Whisper model. In this approach we aim to build ASR model for languages with limited digital resources by sequentially adapting the model across linguistically similar languages. We experimented this on the Malasar language, a Dravidian language spoken by approximately ten thousand people in the Western Ghats of South India. Malasar language faces critical challenges for technological intervention due to its lack of a native script and absence of digital or spoken data resources. Working in collaboration with Wycliffe India and Malasar community members, we created a spoken Malasar corpus paired with transcription in Tamil script, a closely related major language. In our approach to build ASR model for Malasar, we first build an intermediate Tamil ASR, leveraging higher data availability for Tamil annotated speech. This intermediate model is subsequently fine-tuned on Malasar data, allowing for more effective ASR adaptation despite limited resources. The multistage fine-tuning strategy demonstrated significant improvements over direct fine-tuning on Malasar data alone, achieving a word error rate (WER) of 51.9%, which is 4.5% absolute reduction when compared to the direct fine-tuning method. Further a WER reduction to 47.3% was achieved through punctuation removal in post-processing, which addresses formatting inconsistencies that impact evaluation. Our results underscore the effectiveness of sequential multistage fine-tuning combined with targeted post-processing as a scalable strategy for ASR system development in low-resource languages, especially where linguistic similarities can be leveraged to bridge gaps in training data.