Preventing Overdoses and Saving Lives Act of 2021 This bill establishes a grant program for states, Indian tribes, and territories to respond to the opioid crisis. In awarding the grants, the Department of Health and Human Services (HHS) must give priority to recipients with the highest opioid dispensing rates. Recipients must use at least 20% of grant funds to implement a program that requires a prescription for an opioid overdose reversal drug along with an opioid prescription if (1) a patient is prescribed benzodiazepines, medication-assisted treatment, or high-dose opioids; or (2) the prescriber has reason to know of the patient's history with substance use disorders. As part of this program, recipients must also require continuing education on opioid-prescribing practices for prescribers and on recognizing opioid overdose fatalities for coroners. Recipients must use the remaining funds to develop a strategic plan for the jurisdiction's response to the opioid crisis.