Brownian noise of dielectric mirror coatings is expected to be one of the limiting noise sources, at the peak sensitivity, of next generation ground based interferometric gravitational wave (GW) detectors. The use of higher-order Laguerre-Gauss (LG) beams has been suggested to reduce the effect of coating thermal noise in future generations of gravitational wave detectors. In this paper we describe the first test of interferometry with higher-order LG beams in an environment similar to a full-scale gravitational wave detector. We compare the interferometric performance of higher-order LG modes and the fundamental mode beams, injected into a 10 m long suspended cavity that features a finesse of 612, a value chosen to be typical of future gra...