Yes, narcotics can affect blood pressure.
Opioid drugs in part work be depressing the part of the central nervous system. This explains the common side effects of opiates/narcotics (like morphine, oxycodone, fentanyl, hydromorphone, etc) like hypotension - low blood pressure - , respiratory depression (slowing of the breathing rate) - and slowed heart rate.
On the other hand, you should note that pain can significantly raise the blood pressure.
To your health,