We have synthesized previously unreported high-entropy alloys (HEAs) in the pentanary (ScZrNb)_1-x[RhPd]_x and hexanary (ScZrNbTa)_1-x[RhPd]_x systems. The materials have CsCl-type structures and mixed site occupancies. Both HEAs are type-II superconductors with strongly varying critical temperatures (T_c's) depending on the valence electron count (VEC); the T_c's increase monotonically with decreasing VEC within each series, and do not follow the trends seen for either crystalline or amorphous transition metal superconductors. The (ScZrNb)_0.65[RhPd]_0.35 HEA with the highest T_c, ∼9.3 K, also exhibits the largest μ₀H_c2(0) = 10.7 T. The pentanary and hexanary HEAs have higher superconducting transition temperatures than their simple binary intermetallic relatives with the CsCl-type structure and a surprisingly ductile mechanical behavior. The presence of niobium, even at the 20% level, has a positive impact on the T_c. Nevertheless, niobium-free (ScZr)_0.50[RhPd]_0.50, as mother-compound of both superconducting HEAs found here, is itself superconducting, proving that superconductivity is an intrinsic feature of the bulk material.