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 gravitational wave detectors. We found that the expected mode degeneracy of the injected LG(3,3) beam was resolved into a multiple peak structure, and that the cavity length control signal featured several nearby zero crossings. The break up of the mode degeneracy is due to an astigmatism (defined as vertical bar R-cy - R-cx vertical bar) of 5.25 +/- 0.5 cm on one of our cavity mirrors with a radius of curvature (R-c) of 15 m. This observation agrees well with numerical simulations developed with the FINESSE software. We also report on how these higher-order mode beams respond to the misalignment and mode mismatch present in our 10 m cavity. In general we found the LG(3,3) beam to be considerably more susceptible to astigmatism and mode mismatch than a conventional fundamental mode beam. Therefore the potential application of higher-order Laguerre-Gauss beams in future gravitational wave detectors will impose much more stringent requirements on both mode matching and mirror astigmatism.