What if I put the pieces in a baggie?!!?! I'm sure this shear moment of brilliance has not eluded you, but just in case it has...it's simple!
Ziploc type bag
Cut out a picture of the completed puzzle from the box
Put all the pieces of the puzzle in the bag
Tape the picture to the bag
Voila! Now we can store these puzzles in a smaller space, and they can clean them up easily!! :)