Large Language Models (LLMs) have demonstrated remarkable capabilities across a range of Natural Language Processing (NLP) tasks when guided by carefully designed prompts. However, their performance degrades substantially on low-resource languages, where limited pretraining data and the absence of large-scale labeled corpora render conventional prompt engineering strategies ineffective. In this paper, we introduce Evolutionary Prompt Optimization for Low-Resource Language Models (EvoPrompt-LR), a novel framework that employs a hybrid Genetic Algorithm with Simulated Annealing (GA-SA) to automatically discover optimal prompt structures for low-resource languages. EvoPrompt-LR features four key innovations: 1) a hierarchical prompt genome encoding that jointly optimizes instruction templates, in-context example selection and ordering, Chain-of-Thought (CoT) scaffold structures, and output format specifications, 2) a cross-lingual fitness transfer mechanism that evaluates prompt candidates on high-resource proxy languages before selectively fine-evaluating top candidates on the target low-resource language, reducing expensive evaluations by approximately 80%, 3) semantic diversity preservation via embedding-space crowding distance to maintain a diverse population of prompt candidates, and 4) language-specific mutation operators that respect the morphological and syntactic constraints of typologically diverse languages, including agglutinative (Vietnamese, Turkish, Uzbek), fusional (Arabic), and isolating (Khmer) language families. We evaluate EvoPrompt-LR on six low-resource languages Arabic (dialectal), Vietnamese, Swahili, Yoruba, Uzbek, and Khmer across four NLP tasks: Sentiment analysis, Named Entity Recognition (NER), Question Answering (QA), and text summarization, using LLaMA-3-8B, Mistral-7B, and AceGPT-7B as base models. Experimental results demonstrate that EvoPrompt-LR achieves improvements of 8–22% in accuracy and F1 score over the strongest baselines, with the largest gains observed on truly low-resource languages (Yoruba: +22%, Khmer: +19%). On four of six languages, EvoPrompt-LR attains performance competitive with Supervised Fine-Tuning (SFT) approaches that use 100× more labeled data.