The whole world almost came to a standstill due to the COVID-19 pandemic. In response to this pandemic, most psychological responses have been at a higher prevalence such as stress, anxiety, depression, lower mental wellbeing, poor sleep quality, along with the fear of contracting the COVID-19 disease. Related to these psychological responses, previous studies have suggested an association between COVID-19 fear and stress as well as stress and sleep quality. The present study aimed to assess the association between COVID-19 fear and sleep quality and investigate the possible mediating effect of perceived stress during this pandemic. The data were collected, utilizing a self-report online questionnaire, from a sample of 521 Bangladeshi adults. Results showed significant gender differences in COVID-19 fear, stress, and sleep quality and also a significant association between these variables. Mediation results revealed that stress fully mediated the association between COVID-19 fear and sleep quality for males and partially mediated it for females. These results suggest that women are in a more vulnerable position in this pandemic. These findings would be helpful to mental health professionals and policy makers to address the psychological problems induced by the current pandemic.