Opioid Prescription Verification Act of 2021 This bill modifies overdose prevention activities carried out by the Department of Health and Human Services (HHS) and the Centers for Disease Control and Prevention (CDC). Specifically, HHS must update pharmacist training materials to include information about how to verify the identity of patients who are prescribed medications. In addition, the CDC may give preference, in awarding grants for overdose prevention, to states that maintain a prescription drug monitoring program that mandates the collection of specified information, require prescribers of certain potentially addictive medications to issue prescriptions electronically (subject to some exceptions), and require dispensers of certain potentially addictive medications to enter particular information about the purchase of such medications into their state's prescription drug monitoring program.