We report the observation of superconductivity in (Pt0.2Ir0.8)3Zr5 with a chiral space group (P6122) at low temperatures. The bulk nature of the superconductivity at a transition temperature of 2.2 K was confirmed using specific heat measurements. We revealed that (Pt0.2Ir0.8)3Zr5 obeys the weak-coupling Bardeen–Cooper–Schrieffer model, and the dominant mechanism in the upper critical field is the orbital pair-breaking limit rather than the Pauli–Clogston limit. This indicates that the antisymmetric spin–orbit coupling caused by the chiral crystal structure does not significantly affect the superconductivity of (Pt0.2Ir0.8)3Zr5.