It is shown that a supersymmetric and κ-symmetric D3-brane action on AdS₅ × S⁵ is mapped into itself by a duality transformaion, thereby verifying the SL(2,Z) invariance of the D3-brane action in the AdS₅ × S⁵ background as in the flat background. To this end, we fix the κ-symmetry in a gauge which simplifies the classical action in order to perform an SO(2) rotation of the N = 2 spinor index in a manifest way, though this may not be necessary. This situation is the same as the case of a super D-string on AdS₅ × S⁵ where it was shown that the super D-string action is transformed to a form of the IIB Green-Schwarz superstring action with the SL(2,Z) covariant tension in the AdS₅ × S⁵ background through a duality transformation. These results strongly suggest that various duality relations originally found in the flat background may be independent of background geometry, in other words, the duality transformations in string and p-brane theories may exist even in general curved space-time.