Spin-crossover molecules are very appealing for use in multifunctional spintronic devices because of their ability to switch between high-spin and low-spin states with external stimuli such as voltage and light. In actual devices, the molecules are deposited on a substrate, which can modify their properties. However, surprisingly little is known about such molecule-substrate effects. Here we show for the first time, by grazing incidence X-ray diffraction, that an FeII spin-crossover molecular layer displays a well-defined epitaxial relationship with a metal substrate. Then we show, by both density functional calculations and a mechanoelastic model, that the resulting epitaxial strain and the related internal pressure can induce a partial spin conversion at low temperatures, which has indeed been observed experimentally. Our results emphasize the importance of substrate-induced spin state transitions and raise the possibility of exploiting them.