We report on the first detection of very high-energy gamma-ray emission from the Crab Nebula by a Cherenkov telescope in dual-mirror Schwarzschild-Couder (SC) configuration. This result has been achieved by means of the 4 m ASTRI-Horn telescope, operated on Mt. Etna, Italy, and developed in the context of the Cherenkov Telescope Array Observatory preparatory phase. The dual-mirror SC design is aplanatic and characterized by a small plate scale, which allows us to implement large cameras with a large field of view, with small-size pixel sensors and a high level of compactness. The curved focal plane of the ASTRI camera is covered by silicon photo-multipliers, managed by an unconventional front-end electronic system that is based on a customized peak-sensing detector mode. The system includes internal and external calibration systems, hardware and software for control and acquisition, and the complete data archiving and processing chain. These observations of the Crab Nebula were carried out in December 2018 during the telescope verification phase for a total observation time (after data selection) of 24.4 h, equally divided between on- and off-axis source exposure. The camera system was still under commission and its functionality was not yet completely exploited. Furthermore, due to recent eruptions of the Etna Volcano, the mirror reflection efficiency was reduced. Nevertheless, the observations led to the detection of the source with a statistical significance of 5.4σ above an energy threshold of ∼3 TeV. This result provides an important step toward the use of dual-mirror systems in Cherenkov gamma-ray astronomy. A pathfinder mini-array based on nine ASTRI-like telescopes with a large field-of-view is in the course of implementation.