V.J. Katz in History of Topology:
Although Cartan realized in 1899 [1] that the three theorems of vector calculus (Gauss, Green, Stokes) could be easily stated using differential forms, it was Edouard Goursat (1858-1936) who in 1917 [2] first noted that these three theorems were all special cases of a generalised Stokes theorem for differential forms, $$\int_M d\omega=\int_{\partial M}\omega,$$ first stated in coordinate free form by Volterra in 1889 [3].
[1] E. Cartan, Sur certaines expressions différentielles et sur le problème de Pfaff, Ann. École Normale 16 (1899) 230-332.
[2] E. Goursat, Sur certaines systèmes d'équations aux différentielles totales et sur une généralisation du problème de Pfaff, Ann. Fac. Sci. Toulouse (3) 7 (1917), 1-58.
[3] V. Volterra, Sulle funzioni coniugate, Rendiconti Accademia dei Licei (4) 5 (1889), 599-611.
Victor Katz remarks elsewhere that the connection between differential forms and the big three theorems of vector calculus, as expressed by the generalized Stokes theorem, did not appear in textbooks until the second half of the 20th century, the first occurrence probably being in a 1959 Advanced Calculus text.