I have a Tune-o-Matic bridge on my Les Paul and dust loves to accumulate under there, and under the strings between the pickups. It's hard to get a rag under there to clean it without leaving some of the rag behind on the strings.

Is there a better way to clean that area of the guitar?
I know what you mean...and it looks worse if you have a darker guitar
I normally don't worry about it much and just clean it when I change my strings. But, yeah, Q-tip or soft paintbrush.
Just let the good times roll.