We report on the first experimental demonstration of higher-order Laguerre-Gauss (LGpℓ) mode generation and interferometry using a method scalable to the requirements of gravitational wave (GW) detection. GW detectors which use higher-order LGpℓ modes will be less susceptible to mirror thermal noise, which is expected to limit the sensitivity of all currently planned terrestrial detectors. We used a diffractive optic and a mode-cleaner cavity to convert a fundamental LG00 Gaussian beam into an LG33 mode with a purity of 98%. The ratio between the power of the LG00 mode of our laser and the power of the LG33 transmitted by the cavity was 36%. By measuring the transmission of our setup using the LG00, we inferred that the conversion efficiency specific to the LG33 mode was 49%. We illuminated a Michelson interferometer with the LG33 beam and achieved a visibility of 97%.