That's the thing though; the asteroid didn't make the dinosaurs go extinct. Oh sure, it's considered the extinction event; however, it didn't happen overnight. Plus, not all of the dinosaurs went extinct. That's why we still have gators, crocs, and birds; they all descended from dinosaurs that survived the mass climate change brought on by the asteroid.
Now, I'm not sure of the specifics, but I do believe that arthropod evolution was Pre-Cambrian; or at least, close enough to that point to not make much difference. All that would have had to happen is that whatever extinction even occurred in that timeframe to not happen, allowing arthropods to evolve to full sapience. What's really a brain twister is that there's the possibility that this sort of thing did happen, if you accept some of the more out-there theories of how the universe works.