The general objective of this paper was to determine how companies in the olive sector could convert the comparative advantages of olive-growing regions (e.g., culture, tradition, raw materials, knowledge, infrastructure, networks, technological centers, etc.) into competitive advantages, to internationalize, in an accelerated way, and become born global firms, contributing to economic, social, and sustainable development of regions. Thus, we analyzed four cases of exporting companies in this sector (two born global and two non-born global) in southern Spain (Jaén). We chose this province because it is the world’s leading producer of olive oil and, yet it is only the fourth largest exporter compared to the rest of Spain. For the case study, we conducted (and recorded) personal, semi-structured interviews with the founders/managers or individuals in charge of internationalization. To obtain our results, we used a data sheet that included an action protocol, we analyzed each case individually, and we employed sensemaking and pattern-matching techniques to add validity and reliability to the research. Finally, we proposed the “keys” for these companies to go international in an accelerated way, as it would increase their competitiveness, foster the creation of employment, develop networks between companies, boost investment in innovation, etc. The results indicate that it is necessary to follow market orientation, networking, and international entrepreneurship strategies, and that intellectual capital (human, organizational, relational, and technological) of companies (and, therefore, of regions) will be the means through which competitive capabilities are achieved.