We know some people who take the games apart and put them in a big resealable baggie so all the pieces are in one place. You can take this suggestion and modify it so that the parts of puzzles and games are kept in one sandwich size bag, but still kept in the original box.
Or you can pull all the pieces and toys together and put them in clear plastic shoe boxes (which are wonderful for so many things!). This way you child can see what toys or game pieces are in the box! Have your kids make pictures of the items that go in the box and tape it to the end so when they are finished playing they know what bin to use to put away their toys.