The lower critical magnetic field, H-c1, of superconductors is measured by optical magnetometry using ensembles of nitrogen-vacancy centers in diamond. The technique is minimally invasive and allows accurate detection of the vector magnetic field with subgauss sensitivity and submicrometer spatial resolution. These capabilities are used for detailed characterization of the first vortex penetration into superconducting samples from the corners. Aided by the revised calculations of the effective demagnetization factors of actual cuboid-shaped samples, these measurements provide precise determination of H-c1 and the related absolute value of the London penetration depth, lambda. We apply this method to three well-studied superconductors: optimally doped Ba(Fe1-xCox)(2)As-2, stoichiometric CaKFe4As4, and the high-T-c cuprate YBa2Cu3O7-delta. Our results compared well with the values of lambda obtained with other techniques, thus adding another noninvasive and sensitive method to measure these important parameters of superconductors.