We demonstrate the experimental feasibility of a Trojan-horse attack that remains nearly invisible to the single-photon detectors employed in practical quantum key distribution (QKD) systems, such as Clavis2 from ID Quantique. We perform a detailed numerical comparison of the attack performance against Scarani-Ac´ın-Ribordy-Gisin (SARG04) QKD protocol at 1924 nm versus that at 1536 nm. The attack strategy was proposed earlier but found to be unsuccessful at the latter wavelength, as reported in N. Jain et al., New J. Phys. 16, 123030 (2014). However at 1924 nm, we show experimentally that the noise response of the detectors to bright pulses is greatly reduced, and show by modeling that the same attack will succeed. The invisible nature of the attack poses a threat to the security of practical QKD if proper countermeasures are not adopted.