The theme you’re using has to make use of the wp_link_pages() function. Pretty simple to do, but not every theme does it.
This post was written explicitly to test this. You already know whether or not it worked, but for me writing this it’s a big experiment. Drum roll please!
Cool! It works with Twenty Thirteen! (Although the page links are positioned below the sharing buttons, which is mildly annoying to me.) It will be interesting (for me) to see if it still works when I try other themes.