Due to the COVID-19 pandemic, and pursuant to the governments' order of citizens remaining at home, several countries were required to transition from face-to-face instruction to an online model to provide higher education to their students. While factors affecting the use of online learning are diverse and have been studied by models of use and acceptance of technology, this cross-sectional study explores the factors unique to the current emergency situation that influence students' use and acceptance of emergency online learning. Moreover, it proposes a model to predict a student's cognitive engagement in Mexico, Peru, Turkey, and the USA. This is a quantitative study with an exploratory and descriptive scope and cross-sectional design. Data was collected from 1009 students from the four countries, who completed surveys anonymously. The factors analyzed were attitude, affect, and motivation, perceived behavioral control (ease of use, self-efficacy, and accessibility), and cognitive engagement. The data was analyzed using descriptive, correlation, and regression analysis. The predictive model shows that students' attitude toward online learning impacts their cognitive engagement in Mexico, Peru, and the USA. Furthermore, self-efficacy is a significant moderator for cognitive engagement in all four countries. The model also shows that each country has different determinants for cognitive engagement. Understanding the factors that affect the use of emergency online learning is essential for the success and/or achievement of its maximum benefits in situations like a global pandemic. Limitations of this study have been identified as use of convenience sampling, and an inability to explore factors related to instruction and system attributes. Professors who did not teach online learning lacked knowledge about online educational strategies and used the technological resources that were immediately available to them. Therefore, research that explores the use of instructional strategies and the use of technological systems during emergency online learning is necessary. This study includes suggestions to incorporate open educational resources that use microlearning and emphasizes the importance of student self-efficacy; because it was predictor of cognitive engagement in all four countries. Faculty and higher education institutions can and should develop strategies to increase students' sense of self-efficacy.