There are at least three vines native to the U.S. with palmate leaves that are aggressive growers. Please look at these images of native grape (Vitus riparia), Virginia creeper (Parthenocissus quinquefolia), and wild cucumber vine (Echinocystis lobata). Annual wild cucumber vines are especially noted for their ability to climb and engulf small trees and shrubs. (See last image.)




If the vine is wild cucumber, read more here:
http://www.extension.umn.edu/projects/yardandgarden/ygbriefs/h524cuke-wild-bur.html

For control of perennial vines, find the main stems. You can try hand pulling or grubbing (best when the soil is damp); severing the vine and then treating the regrowth with an herbicide such as glyphosate; or applying an herbicide such as triclopyr esther to individual leaflets. You will have to follow the label directions exactly to avoid damage to your lilacs. It may take more than one application of herbicide for good control.

For useful information on using herbicides to kill off undesirable landscape plants, read:
http://www.extension.umn.edu/yardandgarden/YGLNews/YGLN-oct0105.html

It is tempting to pull off the vines, but do so with care. You could just cut the main stems and allow the vines to wilt and remove them later when you will do less damage to the lilacs.