Taking advantage of light-emitting-diode (LED) performance breakthrough driven by the lighting market, we report, to the best of our knowledge, the first LED-pumped chromium-doped crystal laser oscillator and amplifier based on alexandrite crystals (Cr3+:BeAl2O4). We developed a Ce:YAG concentrator as the pumped source, illuminated by blue LEDs that can be easily power scaled. With 2200 LEDs (450 nm), the Ce:YAG concentrator can deliver to the gain medium up to 268 mJ at 10 Hz at 550 nm with a irradiance of 8.5 kW/cm2. We demonstrate, in oscillator configuration, an LED-pumped alexandrite laser delivering an energy of 2.9 mJ at 748 nm in free running operation. In the cavity, we measured a double-pass small signal gain of 1.28, which is in good agreement with numerical simulations. As an amplifier, the system demonstrated to boost a CW Ti:sapphire laser by a factor of 4 at 750 nm in eight passes with a large tuning range from 710 nm to 800 nm.