Asylum seekers are faced with high levels of post-migratory stress due to uncertainty and uncontrollability of the application process, resulting in higher levels of mental health problems. Little is known about the coping strategies utilized by asylum seekers in this context. Structural equation modeling and the stepwise modeling approach were utilized on cross-sectional data from a cohort of asylum seekers in Sweden (N = 455) to examine whether adaptive coping in the form of problem-focused and cognitive-based coping would buffer the impact of post-migratory stressors by moderating the relationship between the stressors and well-being. Fit indices showed good to excellent fit of the final model that regressed well-being on selected post-migratory stressors and coping (CFI = 0.964, RMSEA = 0.043 (90% CI = 0.035–0.051), SRMR = 0.044). Well-being was negatively and significantly regressed on both perceived discrimination (B = −0.42, SE = 0.11, p < 0.001) and distressing family conflicts (B = −0.16, SE = 0.07, p = 0.037), and positively and significantly regressed on cognitive restructuring (B = 0.71, SE = 0.33, p = 0.030). There was, however, no evidence that coping strategies modified the adverse associations between the two post-migratory stressors and well-being. Interventions and policies should prioritize improving contextual factors inherent in the asylum-seeking process in order to reduce stress and enable coping.