Cluck, cluck, Sew came up with the best idea ever when it comes to toddler puzzles, to corral all those stray puzzle pieces that end up all over the house.
She used fleece; I used felt because I have tons of it on hand. She used velcro, and I opted to use snaps and buttons (once again, because I'm running low on velcro, but have dozens of snaps and millions of buttons). Because of the snaps, the button is actually not functional. I just sewed it on to hide the ugly snap.
The way she did it also uses less fabric than the way I did it. She's smarter than me... I should have reread the post before I started! I ended up sewing a large envelope, and then hotgluing it to the back of the puzzle. Either way, it works!