(1) Background. One of the most innovative gamification techniques emerging in the current educational context is the escape room, whose implementation aims to promote student motivation and learning. A review of the literature shows that many of the previous works lack a solid theoretical basis when it comes to explaining the motivational effects associated with student participation in this experience. Therefore, the aim of this study is to analyze the relationship between gamification and motivation and identify which dimensions of gamification experience—through an educational escape room—better predict students’ motivation, based on the framework of self-determination theory. (2) Methods. This research develops a quantitative and correlational methodology. A total of 135 first-year university students participated. (3) Results. The results of the regression models showed that creative thinking, activation, and dominance predicted intrinsic motivation toward knowledge, achievement, and stimulating experiences. Likewise, a higher level of negative affect was predictive of amotivation and external regulation motivation in students. (4) Conclusions. These findings provide new empirical evidence that could guide the design of educational experiences through escape rooms in the context of higher education.