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Uncertainty quantification (UQ) is an active area of research, and an essential technique used in all fields of science and engineering. The most common methods for UQ are Monte Carlo and surrogate-modelling. The former method is dimensionality independent but has slow convergence, while the latter method has been shown to yield large computational speedups with respect to Monte Carlo. However, surrogate models suffer from the so-called curse of dimensionality, and become costly to train for high-dimensional problems, where UQ might become computationally prohibitive. In this paper we present a new technique, Lasso Monte Carlo (LMC), which combines a Lasso surrogate model with the multifidelity Monte Carlo technique, in order to perform UQ in high-dimensional settings, at a reduced computational cost. We provide mathematical guarantees for the unbiasedness of the method, and show that LMC can be more accurate than simple Monte Carlo. The theory is numerically tested with benchmarks on toy problems, as well as on a real example of UQ from the field of nuclear engineering. In all presented examples LMC is more accurate than simple Monte Carlo and other multifidelity methods. Thanks to LMC, computational costs are reduced by more than a factor of 5 with respect to simple MC, in relevant cases.