The light-quark sector of the Λ(1405) baryon is examined in the context of the recent discovery of a dominant antikaon-nucleon composition at low quark masses. Further evidence for this interpre- tation of the Λ(1405) is presented, by calculating the u and d quark contributions to the Λ(1405) magnetic form factors in lattice QCD. The extent to which these quantities are consistent with the exotic molecular description can then be quantified by comparing the results with the equiv- alent nucleon form factors. Drawing on a recent extension of the graded-symmetry approach for the flavor-singlet components of the Λ(1405), the separation of the connected and disconnected contributions is performed in both the flavor-octet and singlet representations. In both cases, the disconnected loop contributions are found to be unexpectedly large. The relationship between the light-quark contributions to the Λ(1405) magnetic form factor and the connected contributions of the nucleon magnetic form factors is thus confirmed in the case of lattice QCD, establishing compelling evidence for a KN molecular structure of the Λ(1405) near the physical point.