Ask Mr. Smarty Plants

Campsis radicans (Trumpet creeper) is a native, colorful flowering vine, growing natively through most of North America, but most invasive in the Southeast. Follow the plant link to learn about the risks of having the plant in your yard. 

Also, Dave's Garden, which is a forum, has 64 negative comments on the plant.

From Floridata, more comments on invasiveness of Campsis radicans.

Previous Mr. Smarty Plants answer

Conclusion: How to keep it from being invasive? Don't plant it and don't let it into your garden from anywhere else. It can definitely harm your pecan tree if it gets up in it and covers the leaves, preventing sunlight from reaching those leaves and also preventing photosynthesis, whereby the plant uses the energy of the sun to produce food for the plant as well as oxygen for our air.

Obviously, you already have it. We can pass along some of the suggestions for controlling it. First; patience. Even if you never get rid of it completely, it will only get worse if you don't stay after it year after year. Second: herbicide. Do NOT spray herbicide, this will only damage the tree, other desirable plants in your garden and the environment, but won't get close to all those Trumpet Vine roots underground. You say it's in your lawn-mow it, low and regularly. Mowing won't kill it, but it will slow it down. Get a bottle of an undiluted wide spectrum herbicide and some disposable sponge brushes. With garden nippers, clip off the stems close to the ground and immediately paint the cut edge of the stem in hopes you get it into the system of the vine before the cut place heals over in self defense. This makes it possible for the herbicide to actually get to the roots. Sometimes. If you have big roots going up a tree, by all means, pull them away, cut the vine and, again, paint the fresh cut with the herbicide.

The only way to keep a plant from being invasive is to never plant it.