When a quantum dot is attached to a metallic reservoir and a superconducting contact, Andreev processes lead to a finite subgap current at the normal lead and the creation or destruction of Cooper pairs. Andreev reflection engines profit from the destruction of Cooper pairs to provide the work needed to set a charge current at the normal-conductor contact generating electrical power. For this power-transduction device, high power and large efficiencies in quantum mechanically enhanced regimes are demonstrated. There thermodynamic tradeoff relations between power, efficiency, and stability, valid for any classical engine, are overcome, and kinetic constraints on the engine precision are largely surpassed in arbitrary far-from-equilibrium conditions.