Magnetism and spin-orbit coupling are two quintessential ingredients underlying topological transport phenomena in itinerant ferromagnets. When spin-polarized bands support nodal points/lines with band degeneracy that can be lifted by spin-orbit coupling, the nodal structures become a source of Berry curvature, leading to a large anomalous Hall effect. However, two-dimensional systems can possess stable nodal structures only when proper crystalline symmetry exists. Here we show that two-dimensional spin-polarized band structures of perovskite oxides generally support symmetry-protected nodal lines and points that govern both the sign and the magnitude of the anomalous Hall effect. To demonstrate this, we performed angle-resolved photoemission studies of ultrathin films of SrRuO₃, a representative metallic ferromagnet with spin-orbit coupling. We show that the sign-changing anomalous Hall effect upon variation in the film thickness, magnetization and chemical potential can be well explained by theoretical models. Our work may facilitate new switchable devices based on ferromagnetic ultrathin films.