© 2016 American Physical Society. Ferromagnetic insulating La2CoMnO6-É (LCMO) epitaxial thin films grown on top of SrTiO3 (001) substrates present a strong magnetic anisotropy favoring the out-of-plane (OP) orientation of the magnetization with a large anisotropy field (∼70 kOe for film thickness of about 15 nm). Diminishing oxygen off-stoichiometry of the film enhances the anisotropy. We attribute this to the concomitant shrinkage of the OP cell parameter and to the increasing of the tensile strain of the films. Consistently, LCMO films grown on (LaAlO3)0.3(Sr2AlTaO6)0.7 and LaAlO3 substrates (with a larger OP lattice parameter and compressive stress) display in-plane (IP) magnetic anisotropy. Thus, we link the strong magnetic anisotropy observed in LCMO to the film stress: tensile strain favors perpendicular anisotropy, and compressive stress favors IP anisotropy. We also report on the thickness dependence of the magnetic properties. Perpendicular anisotropy, saturation magnetization, and Curie temperature are maintained over a large range of film thickness.