The present article explores the impact of educational robotics on fostering computational thinking and problem-solving skills in elementary school students through a problem-based learning approach. This study involved the creation of a framework which includes a robot and two eBooks designed for students and teachers. The eBooks serve as a guide to the construction and programming of a small Arduino-based robot. Through integration with gamification elements, the model features a narrative with three characters to boost a student’s engagement and motivation. Through iteration of heuristic evaluations and practical tests, we refined the initial theoretical framework. An empirical study was conducted in two phases involving 350 students. The first empirical test involved a small group of 21 students, similar to end users, from five European schools. With a 100% completion rate for the tasks, 73.47% of these tasks were solved optimally. Later, we conducted a larger validation study which involved 329 students in a Portuguese school. This second phase of the study was conducted during the 2022–2023 and 2023–2024 school years with three study groups. The results led to a 91.13% success rate in problem-solving activities, and 56.99% of those students achieved optimal solutions. Advanced statistical techniques, including ANOVA, were applied to account for group differences and ensure the robustness of the findings. This study demonstrates that the proposed model which integrates educational robotics with problem-based learning effectively promotes computational thinking and problem-solving skills, which are essential for the 21st century. These findings support the inclusion of robotics into primary school curricula and provide a validated framework for educators.