Employees at healthcare organizations are experiencing more stress than ever given the current COVID-19 pandemic. Different types of stress are affecting diverse organizational outcomes, including the employees’ voluntary turnover. This is the case of cognitive stress, a type of stress that affects how individuals process information, which can influence employees’ turnover intentions. In this study, we look at the mechanisms that can reduce the adverse effects of cognitive stress on turnover intentions, particularly the role of employees’ perceived psychological safety (i.e., how safe they perceive the interactions with their colleagues are). We hypothesize that psychological safety mediates the relationship between cognitive stress and turnover intentions, and COVID-19 worry and supervisor support moderate the relationship between cognitive stress and psychological safety. To test our hypothesis, we invited two public health care organizations in Chile to join this study. In total, we obtained a sample of 146 employees in 21 different teams. Using a multilevel model, we found that psychological safety prevents the harmful effects of cognitive stress on employees’ turnover intentions. In addition, while COVID-19 worry can worsen the relationship between cognitive stress and psychological safety, supervisor support only directly affects psychological safety. This study contributes to expanding the stress and psychological safety literature and informs practitioners in healthcare organizations about how to deal with cognitive stress in the “new normality” that the pandemic has brought.