Authenticity is an important topic in business activities and society at large. Sustainability management and marketing activities are required as consumers become more interested in society. The consumer evaluation of brand authenticity is a crucial factor in a brand’s reputation and long-term growth, which affects brand loyalty, electronic word of mouth (eWOM) and purchase behavior. With this latest trend, luxury brands are paying attention to sustainability activities, but consumer confidence in the sustainability of luxury brands is still low. Experts emphasize that building trust with customers by promising authenticity is a key challenge. Meanwhile, luxury brands are actively communicating with consumers through SNS, such as Instagram. Social media marketing directly and indirectly influences brand authenticity, as well as purchase intentions and eWOM. Nevertheless, there is little exposure to sustainability activities in the social media accounts of luxury brands. Therefore, the purpose of this study is to examine the relationship between the sustainability activities on the social media of luxury brands and brand authenticity, as well as the effects of brand authenticity on eWOM and purchase intention. For this study, we conducted an online survey targeting individuals with ages of 20–30 years and who are highly interested in society while actively using social media. The collected data were analyzed using the SPSS 21.0 software package. The main finding of this study reveals that consumers are more aware of brand authenticity when they see an Instagram post that includes messages about sustainability rather than a post without any such messages. In addition, brand authenticity has positive effects on eWOM and purchase intention. These findings suggest that sustainability marketing by luxury brands leads to positive consumer response, and the broadcasting of messages about sustainability via social media is effective for the marketing strategy of luxury brands.