What weed controls won't hurt my evergreen shurbs?

I used to have a flower bed in front of my house. It was a holdover from the previous owners, and because I didn’t have the time, energy, or desire to maintain a big patch of fertile dirt, it became a weed-bed in no time.

So, being the practical sort of guy that I am, I dug it all out in the spring of last year. I removed as much dirt as I could, poured in new topsoil, and planted some arborvitae shrubs. Then I laid down some of that weed barrier cloth and covered it all in decorative gravel. I was proud of my low-maintenance solution, and it looks a lot better anyway. For the rest of the season, I had zero problem with weeds.

This year is a little different. Some weird, creeping crap has infiltrated my formerly sterile environment. Whatever this stuff is (some variant of chickweed is the closest I can guess using some online photos), it seems just happy to grow on and in the gravel, because it’s not actually taking root in the dirt. This means it’s very easy to pull, but it’s so quick to grow that I can’t keep up with it.

I’d like to put down some weed preventative, like Preen. I also might like to use a weed control spray over the area to kill anything that’s germinated but isn’t visible or pullable. However, I don’t know what products out there might harm the shrubs. A lot of lawn weedkillers, for example, distiguish between grasses and broadleaf weeds. But I don’t know which my arborvitaes are more closely related to from a biochemical standpoint.

So, in broad terms, what types of products should and should I not apply to the soil (or gravel) near my shrubs? Is Weed-B-Gon going to think my shrubs are weeds? I don’t need to dig a half-dozed dead trees out of my yard.