Angle-resolved photoemission spectroscopy reveals very surprising strain-induced effects on the electronic band dispersion of epitaxial La2-xSrxCuO4-delta thin films. In strained films we measure a band that crosses the Fermi level (E-F) well before the Brillouin zone boundary. This is in contrast to the flat band reported in unstrained single crystals and in our unstrained films, as well as in contrast to the band flattening predicted by band structure calculations for in-plane compressive strain. In spite of the density of states reduction near E-F, the critical temperature increases in strained films with respect to unstrained samples. These results require a radical departure from commonly accepted notions about strain effects on high temperature superconductors, with possible general repercussions on superconductivity theory.