Open your console and do the following, it will delete the cookies and cache on the deepest part of your computer, therefore deleting old games.
Just write this (without brackets) and your computer should restart, just wait until it runs up again and enjoy, there will be no more old games listed.
Its a little long process but chess.com wanted it to be the hard way, there is no other way around.
[don't post potentially harmful instructions- - MOD]
How do I delete old games that are listed?