We present a novel statistic to extract cosmological information in weak lensing data: the lensing minima. We also investigate the effect of baryons on cosmological constraints from peak and minimum counts. Using the MassiveNuS simulations, we find that lensing minima are sensitive to non-Gaussian cosmological information and are complementary to the lensing power spectrum and peak counts. For an LSST-like survey, we obtain 95 per cent credible intervals from a combination of lensing minima and peaks that are significantly stronger than from the power spectrum alone, by 44 per cent, 11 per cent, and 63 per cent for the neutrino mass sum ∑m_ν, matter density Ω_m, and amplitude of fluctuation A_s, respectively. We explore the effect of baryonic processes on lensing minima and peaks using the hydrodynamical simulations BAHAMAS and Osato15. We find that ignoring baryonic effects would lead to strong (≍4σ) biases in inferences from peak counts, but negligible (≍0.5σ) for minimum counts, suggesting lensing minima are a potentially more robust tool against baryonic effects. Finally, we demonstrate that the biases can in principle be mitigated without significantly degrading cosmological constraints when we model and marginalize the baryonic effects....