Opioid drugs have analgesic properties used to treat chronic and post-surgical pain due to descending pain modulation. The use of opioids is often associated with adverse effects or clinical issues. This study aimed to evaluate the toxicity of opioids by exposing the neuroblastoma cell line (SH-SY5Y) to 0, 1, 10, and 100 µM oxycodone and naloxone for 24 h. Analyses were carried out to evaluate cell cytotoxicity, identification of cell death, DNA damage, superoxide dismutase (SOD), glutathione S-transferase (GST), and acetylcholinesterase (AChE) activities, in addition to molecular docking. Oxycodone and naloxone exposure did not alter the SH-SY5Y cell viability. The exposure to 100 µM oxycodone and naloxone significantly increased the cells’ DNA damage score compared to the control group. Naloxone exposure significantly inhibited AChE, GST, and SOD activities, while oxycodone did not alter these enzymes’ activities. Molecular docking showed that naloxone and oxycodone interact with different amino acids in the studied enzymes, which may explain the differences in enzymatic inhibition. Naloxone altered the antioxidant defenses of SH-SY5Y cells, which may have caused DNA damage 24 h after the exposure. On the other hand, more studies are necessary to explain how oxycodone causes DNA damage.