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We consider a multivariate finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates and the size of the response may be much larger than the sample size. We provide an ℓ1-oracle inequality satisfied by the Lasso estimator according to the Kullback−Leibler loss. This result is an extension of the ℓ1-oracle inequality established by Meynet in [ESAIM: PS 17 (2013) 650–671]. in the multivariate case. We focus on the Lasso for its ℓ1-regularization properties rather than for the variable selection procedure