Tool For Stripped Screws?

I spend most of my days fixing phones. I find a lot of them have stripped screws.

Does anyone have any advice on a tool i can buy to help me out?

I purchased a dremel tool and micro steel drill bits, but they just seems to drill into the screw a little bit and not enough to actually pop the head off.

Anyone have any tools to recommend? Bearing in mind im working in tiny spaces with mini and most likely flush screws.