Any foreign substance that you put in your body and that your body is not used to make cause someone to fall asleep. Like any drug, they almost always have some sort of side effects. Regarding analgesic drugs, they are no different. Some of the symptoms include rashes, itching, diarrhea, headache, dizziness, breathing problems, swelling, hoarseness and upset stomach.
Some people who take aspirin complain about heartburn and nausea. It really just depends on the analgesic that you take because there are different side effects to the different drugs. Therefore, these side effects alone may make someone sleepy. However, there are certain analgesics that also have a side effect of being drowsy. If this is the case, make sure that you do not drive after taking these.