When I wrote that Wikipedia paragraph, I think I had in mind Theorem 5.16 in Evans & Gariepy, _Measure Theory and the Fine Properties of Functions_, which essentially proves Stokes' theorem for level sets of functions with bounded variation.  Unfortunately, the proof given in the book is just citations to Lemmas 5.2 and 5.5, Theorem 5.15, and "the foregoing theory", so I can't be confident (via a cursory review) that "weak formulation+coarea" is the correct summary.