We present a methodology based on ex situ (postgrowth) electrochemistry to control the oxygen concentration in thin films of the superconducting oxide La2CuO4+y grown epitaxially on substrates of isostructural LaSrAlO4. The superconducting transition temperature, which depends on the oxygen concentration, can be tuned by adjusting the pH level of the base solution used for the electrochemical reaction. As our main finding, we demonstrate that the dopant oxygens can either occupy the interstitial layer in an orientationally disordered state or organize into a crystalline phase via a mechanism in which dopant oxygens are inserted into the substrate, changing the lattice symmetry of both the substrate and the epitaxial film. We discuss this mechanism, and we describe the resulting methodology as a platform to be explored in thin films of other transition-metal oxides.