Code-mixed languages, a mix of native languages with English, is increasingly prevalent in Indian social media communications. Sentiment analysis in Indian code-mixed languages faces significant challenges due to complex grammar structures, diverse and non-standard phonetic spellings, and scarcity of annotated corpora. This paper presents INSAG, a novel Agentic AI system that employs specialized autonomous agents, each leveraging large language models (LLMs), to manage subtasks such as transliteration, word-level translation, phonetic normalization, and sentiment scoring. A central orchestration agent dynamically coordinates these modular agents and synthesizes their outputs to determine the final sentiment polarity. Additionally, we propose a phonetic embedding technique to manage orthographic inconsistencies arising from the Romanized representation of Indian languages. An experimental study on two publicly available code-mixed datasets, SemEval 2020 Task 9 (Hindi-English) and BnSentMix (Bengali-English), demonstrate INSAG’s superior performance, achieving F1 scores of 0.76 and 0.77 respectively, significantly outperforming traditional and transformer-based baselines.