Intrinsic magnetoelectric coupling describes the interaction between magnetic and electric polarization through an inherent microscopic mechanism in a single-phase material. This phenomenon has the potential to control the magnetic state of a material with an electric field, an enticing prospect for device engineering. Here, we demonstrate 'giant' magnetoelectric cross-field control in a tetravalent titanate film. In bulk form, EuTiO 3, is antiferromagnetic. However, both anti and ferromagnetic interactions coexist between different nearest europium neighbours. In thin epitaxial films, strain was used to alter the relative strength of the magnetic exchange constants. We not only show that moderate biaxial compression precipitates local magnetic competition, but also demonstrate that the application of an electric field at this strain condition switches the magnetic ground state. Using first-principles density functional theory, we resolve the underlying microscopic mechanism resulting in G-type magnetic order and illustrate how it is responsible for the 'giant' magnetoelectric effect.