Will Botox Work for Lines Under the Eyes?

If the lines are present at rest, as in your situation, then Botox may not help that much, compared with resurfacing for photodamage-induced lines. This can be done by fractional Fraxel laser or chemical peels. Fillers may look lumpy in this area or may be visible because of the thin skin. Some patients who do have lines underneath the lower eyelids when smiling can obtain a nice result with very small units of Botox as long as there is no tendency for puffiness (the lack of muscle movement after Botox could minimize the draining of fluids and if one is prone to puffiness of the lower lids, then it could be compounded). Also, some patients have poor skin tone of the lower eyelid and the lid can be made to drop if Botox is injected there. That can create a dry feeling of the eye.

The other answers are all correct just a couple points:
1. I do think its worth a try with Botox to get rid of the most laterally located lower wrinkles...I definitely have patients for whom this has worked well. The more midline wrinkles aren't appropriate for Botox
2. I would do the Botox first before doing the filler. You may find you need less filler once the resting muscle tone is diminished by the Botox.
My two cents.