...this isn't very correct

Opioids are certainly considered very potent analgesics because of their interactions with the μ, κ, and δ receptors. These generally work through inducing secondary messengers like cAMP to block calcium channels and activate potassium channels, which usually makes it much more difficult for pain sensing neurons to transmit signals. The underlying pathology of addiction is related to dopamine release in the addiction centers like the VTA, but that's certainly different from how pain is actually blocked.

Tylenol is also not a particularly strong analgesic by any stretch, but it works by inhibiting inflammatory pathways. It's often given in conjunction with opioids because it can reduce swelling and discomfort that isn't strictly pain.