Tools from transcendence theory have been crucial to the most significant recent advances in problems of unlikely intersections.  The strategy was first dreamed up by Zannier, I believe, and has been applied with great success by Pila and Habegger, for example here:
http://arxiv.org/abs/1409.0771
The authors prove some major cases of the the Zilber-Pink conjecture, some of them unconditionally, some conditional on conjectures in transcendence theory.  For people who don't know, the Z-P conjecture is an extremely general and very strong statement in diophantine geometry.  As an example of it's strength, Pink has given a *very* short argument that reproves Faltings's Theorem assuming only a very special case of the Z-P Conjecture.