We report on the first experimental demonstration of higher-order Laguerre-Gauss (LG(p)(ℓ)) mode generation and interferometry using a method scalable to the requirements of gravitational wave (GW) detection. GW detectors which use higher-order LG(p)(ℓ) 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 LG(0)(0) Gaussian beam into an LG(3)(3) mode with a purity of 98%. The ratio between the power of the LG(0)(0) mode of our laser and the power of the LG(3)(3) transmitted by the cavity was 36%. By measuring the transmission of our setup using the LG(0)(0), we inferred that the conversion efficiency specific to the LG(3)(3) mode was 49%. We illuminated a Michelson interferometer with the LG(3)(3) beam and achieved a visibility of 97%.