Semiclassical Einstein-Brillouin-Keller (EBK) quantization of the nonintegrabIe Henon-Heiles Hamiltonian succeeds using the Fourier transform method of Martens and Ezra. Two innovations are required for this success : (1) the use of tunneling corrected quantizing actions obtained from an approximate, one-dimensional Hamiltonian and (2) exploitation of intermediate-time approximate quasiperiodicity or "vague tori" wherein the Fourier transform of chaotic motion over 10-100 vibrational periods allows the determination of frequencies and amplitudes which approximate motion during the time interval. Approximate tori, actions, and EBK energy levels are then straightforward. We use an interpolation method to smooth over small resonance zones that are not expected to be important quantum mechanically.