A diversity of predators succeed in overcoming prey with a high-speed pursuit or a high-acceleration strike during an ambush. However, many predatory fishes approach evasive prey at relatively slow speed. One such example is the red lionfish (Pterois volitans). A slow and highly decorated suction feeder, red lionfish successfully stalk evasive prey and have earned a reputation as a formidable and widely invasive predator in the western Atlantic Ocean. To understand the strategy of such slow predators, we performed kinematic measurements of individual red lionfish as they pursued green chromis (Chromis viridis). Despite moving half as fast as their prey, red lionfish successfully captured the chromis in 62% of our experiments, which were performed in a cylindrical arena (1.4 m diameter) for up to one hour. The red lionfish ceaselessly pursued chromis, which avoided the predator with routine swimming. By analyzing the trajectories of predator and prey, we found that red lionfish employed a pure-pursuit strategy, with a heading that was consistently directed toward the prey’s position. The red lionfish did not attempt a predatory strike until the rare instant when prey permitted a close approach (<10 cm). At this distance, red lionfish rarely chose not to strike (<16%) and were usually successful (74% of strikes) in capturing the prey. Prey orientation and fish speed had little influence on a strike or capture. Therefore, the slow red lionfish employs a persistent pure-pursuit strategy that succeeds through chance events when a prey fails to maintain avoidance swimming and falls within striking distance. This persistent strategy may be common in fishes and other animals that pursue faster and more maneuverable prey.