This study investigates the relationship between reactive agility and reaction speed, sprint speed, and muscle strength and their contribution to Y-shaped agility test and reactive agility test (RAT) performance in handball players. Seven handball players performed a Y-shaped agility test and RAT, simple and choice-based reaction time test, reaction test to fast and slow generated visual stimuli, 5 m and 20 m sprint tests, a 505 Agility test, a squat jump, a countermovement jump, and a drop jump. The results reveal a significant relationship between the times for the Y-shaped agility test and the 20 m sprint (r = 0.777, p = 0.040, R = 0.604), as well as the time for the 505 Agility test (r = 0.770, p = 0.043, R2 = 0.593), and RSI (r = −0.755, p = 0.050, R2 = 0.570); however, no relationship was found with parameters pertaining to reaction speed. RAT performance did not significantly correlate with RSI, time in the 505 Agility test, squat jump height, time in the 20 m sprint, and reaction time to fast generated visual stimuli, although large effect sizes were found (r > 0.5). It seems that the contribution of sensory and motor components depends on the structure of the agility test. While the Y-shaped agility test is mainly determined by sprinting, the change of direction speed, and reactive strength, there is the tendency for the reactive agility test to be determined by strength, speed, and visual reaction time.