Introduction: Recent studies show that neuropsychiatric disorders are the most frequent sequelae of COVID-19 in children. Purpose: Our work aimed to evaluate the impact of SARS-CoV-2 infection on behavior and sleep in children and adolescents. Materials and Methods: We enrolled 107 patients aged 1.5–18 years who contracted COVID-19 between one year and one month prior to data collection, referred to the University of Campania Luigi Vanvitelli in Italy. We asked their parents to complete two standardized questionnaires for the assessment of behavior (Child Behavior CheckList (CBCL)) and sleep (Sleep Disturbance Scale for Children (SLDS)). We analysed and compared the results with a control group (pre-COVID-19 pandemic). Results: In the COVID-19 group, the major results were found for sleep breathing disorders, sleep–wake transition disorders and disorders of initiating and maintaining sleep for the SDSC questionnaire, and internalizing scale, total scale and anxiety/depression for the CBCL questionnaire. The comparison of the CBCL results of the cases with the controls revealed statistically significant differences for the following items: internalizing scale, externalizing scale, somatic complaints, total score, thought problems [(p < 0.01)], anxious/depressed problems and withdrawn [(p < 0.001)]. Conclusions: COVID-19 has impacted children’s and adolescents’ mental health. Adolescents were the most affected patient group for internalizing problems, including anxiety and depression.