Near-field scanning optical microscopy is a powerful technique for imaging below the diffraction limit, which has been extensively used in biomedical imaging and nanophotonics. However, when the electromagnetic fields under measurement are strongly confined, they can be heavily perturbed by the presence of the near-field probe itself. Here, taking inspiration from scattering-cancellation invisibility cloaks, Huygens-Kerker scatterers, and cloaked sensors, we design and fabricate a cloaked near-field probe. We show that, by suitably nanostructuring the probe, its electric and magnetic polarizabilities can be controlled and balanced. As a result, probe-induced perturbations can be largely suppressed, effectively cloaking the near-field probe without preventing its ability to measure. We experimentally demonstrate the cloaking effect by comparing the interaction of conventional and nanostructured probes with a representative nanophotonic structure, namely, a 1D photonic-crystal cavity. Our results show that, by engineering the structure of the probe, one can systematically control its back action on the resonant fields of the sample and decrease the perturbation by >70% with most of our modified probes, and by up to 1 order of magnitude for the best probe, at probe-sample distances of 100 nm. Our work paves the way for non-invasive near-field optical microscopy of classical and quantum nanosystems.