Free-space optical communication using the orbital angular momentum (OAM) of light has garnered significant interest lately due to the potentially vast bandwidth intrinsic to the infinite Hilbert space of OAM modes. Unfortunately, OAM light beams suffer from serious distortions due to atmospheric turbulence (AT) that has become a dominant factor limiting the advance of OAM-based free-space communication. Here we propose and demonstrate a free-space communication scheme---using OAM beams and their azimuthal-mode phase-shift for keying (OAM-APSK)---which is resilient to AT-induced distortions. Combined with a digital holographic mode sorting (DHMS) technique, the proposed approach is able to achieve high signal-to-noise ratios and to maintain low modal crosstalk, even for extremely strong turbulence conditions, with magnitudes significantly beyond existing AT mitigation methods. The demonstrated OAM-APSK and DHMS schemes may now open up a great avenue for OAM-based free-space optical communication that could elegantly resolve the long-standing challenge imposed by atmospheric turbulence.