It is challenging to generate high-quality instruction datasets for non-English languages due to tail phenomena, which limit performance on less frequently observed data. To mitigate this issue, we propose translating existing high-quality English instruction datasets as a solution, emphasizing the need for complete and instruction-aware translations to maintain the inherent attributes of these datasets. We claim that fine-tuning LLMs with datasets translated in this way can improve their performance in the target language. To this end, we introduces a new translation framework tailored for instruction datasets, named InstaTrans (INSTruction-Aware TRANSlation). Through extensive experiments, we demonstrate the superiority of InstaTrans over other competitors in terms of completeness and instruction-awareness of translation, highlighting its potential to broaden the accessibility of LLMs across diverse languages at a relatively low cost. Furthermore, we have validated that fine-tuning LLMs with datasets translated by InstaTrans can effectively improve their performance in the target language.