Layer-by-layer epitaxial growth of the pyrochlore magnet Tb2Ti2O7 on the isostructural substrate Y2Ti2O7 results in high-quality single crystal films of up to 60 nm thickness. Substrate-induced strain is shown to act as a strong and controlled perturbation to the exotic magnetism of Tb2Ti2O7, opening up the general prospect of strain-engineering the diverse magnetic and electrical properties of pyrochlore oxides.