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Observations of the redshifted 21-cm signal emitted by neutral hydrogen represent a promising probe of large-scale structure in the universe. However, cosmological 21-cm signal is challenging to observe due to astrophysical foregrounds which are several orders of magnitude brighter. Traditional linear foreground removal methods can optimally remove foregrounds for a known telescope response but are sensitive to telescope systematic errors such as antenna gain and delay errors, leaving foreground contamination in the recovered signal. Non-linear methods such as principal component analysis, on the other hand, have been used successfully for foreground removal, but they lead to signal loss that is difficult to characterize and requires careful analysis. In this paper, we present a systematics-robust foreground removal technique which combines both linear and non-linear methods. We first obtain signal and foreground estimates using a linear filter. Under the assumption that the signal estimate is contaminated by foreground residuals induced by parameterizable systematic effects, we infer the systematics-induced contamination by cross-correlating the initial signal and foreground estimates. Correcting for the inferred error, we are able to subtract foreground contamination from the linearly filtered signal up to the first order in the amplitude of the telescope systematics. In simulations of an interferometric 21-cm survey, our algorithm removes foreground leakage induced by complex gain errors by one to two orders of magnitude in the power spectrum. Our technique thus eases the requirements on telescope characterization for modern and next-generation 21-cm cosmology experiments.