There really isn't a problem with having 100,000 404 "errors." Google's telling you that it thinks 100,000 pages exist, but when it tries to find them, it's getting a 404 code. That's fine: 404s tell Google that a page doesn't exist and to remove the page from Google's index. That's what we want.
The real problem is with your site migration, as FCBM pointed out. If you properly 301 redirect old pages to new, Google will be redirected to the new page, it won't just hit a 404. If you fix the problems with the site migration (not focusing on Google too much), the 404 errors will naturally subside.
The other option is to just take the hit from the migration, and Google will eventually remove all of these pages from its index and stop reporting on them, as long as there aren't live links pointing to the removed pages.
Good luck!