ABSTRACT High‐dimensional regression problems, for example with genomic or drug exposure data, typically involve automated selection of a sparse set of regressors. Penalized regression methods like the LASSO can deliver a family of candidate sparse models. To select one, there are criteria balancing log‐likelihood and model size, the most common being AIC and BIC. These two methods do not take into account the implicit multiple testing performed when selecting variables in a high‐dimensional regression, which makes them too liberal. We propose the extended AIC (EAIC), a new information criterion for sparse model selection in high‐dimensional regressions. It allows for asymptotic FWER control when the candidate regressors are independent. It is based on a simple formula involving model log‐likelihood, model size, the total number of candidate regressors, and the FWER target. In a simulation study over a wide range of linear and logistic regression settings, we assessed the variable selection performance of the EAIC and of other information criteria (including some that also use the number of candidate regressors: mBIC, mAIC, and EBIC) in conjunction with the LASSO. Our method controls the FWER in nearly all settings, in contrast to the AIC and BIC, which produce many false positives. We also illustrate it for the automated signal detection of adverse drug reactions on the French pharmacovigilance spontaneous reporting database.