No one can be entirely certain, but most studies point in the direction of "yes", for instance
The testing, conducted by the US Food and Drug Administration (FDA), ultimately covered more than 100 drugs, prescription and over-the-counter. The results showed that about 90% of them were safe and effective as far as 15 years past their original expiration date.
For a more recent study of the same issue see Cantrell et al.
The Shelf-Life Extension Program (SLEP) checks long-term stability of federal drug stockpiles. Eighty-eight percent of 122 different drugs stored under ideal environmental conditions had their expiration dates extended more than 1 year, with an average extension of 66 months and a maximum extension of 278 months.
In our data set, 12 of 14 medications retained full potency for at least 336 months, and 8 of these for at least 480 months. Given our inability to confirm ideal storage conditions for our samples, our results support the effectiveness of broadly extending expiration dates for many drugs, the efficacy of which has been demonstrated by SLEP in a more controlled fashion.
One (exceptional) fact to note in the latter paper is that aspirin withstood really poorly the test of time: out of 226mg declared, only 1.5mg remained, but this was after a really long time; all drugs tested by Cantrell et al. had expired "28 to 40 years prior to analysis". The first article I mentioned has a more optimistic take on recently expired aspirin:
However, Chris Allen, a vice president at the Bayer unit that makes aspirin, said the dating is "pretty conservative"; when Bayer has tested 4-year-old aspirin, it remained 100% effective, he said. So why doesn't Bayer set a 4-year expiration date? Because the company often changes packaging, and it undertakes "continuous improvement programs," Mr. Allen said. Each change triggers a need for more expiration-date testing, and testing each time for a 4-year life would be impractical. Bayer has never tested aspirin beyond 4 years, Mr. Allen said. But Jens Carstensen has. Dr. Carstensen, professor emeritus at the University of Wisconsin's pharmacy school, who wrote what is considered the main text on drug stability, said, "I did a study of different aspirins, and after 5 years, Bayer was still excellent. Aspirin, if made correctly, is very stable.