An old and well-tested trick is to use glue. True, with multiple screws it will be tedious, and eat up a handful of screwdrivers. But you're going to have to replace the screws anyway, unless this is a one-shot to get at something inside that you want out.
In case you think I'm joking -- I'm not. You're going to glue the tip of the screwdriver to the screw's head, which will then turn.