The use of nanomaterials for strain sensors has attracted attention due to their unique electromechanical properties. However, nanomaterials have yet to overcome many technological obstacles and thus are not yet the preferred material for strain sensors. In this work, we investigated graphene woven fabrics (GWFs) for strain sensing. Different than graphene films, GWFs undergo significant changes in their polycrystalline structures along with high-density crack formation and propagation mechanically deformed. The electrical resistance of GWFs increases exponentially with tensile strain with gauge factors of ~10(3) under 2~6% strains and ~10(6) under higher strains that are the highest thus far reported, due to its woven mesh configuration and fracture behavior, making it an ideal structure for sensing tensile deformation by changes in strain. The main mechanism is investigated, resulting in a theoretical model that predicts very well the observed behavior.