We report on ab initio no-core shell model calculations of the mirror Λ hypernuclei _{Λ}^{4}H and _{Λ}^{4}He, using the Bonn-Jülich leading-order chiral effective field theory hyperon-nucleon potentials plus a charge symmetry breaking Λ-Σ^{0} mixing vertex. In addition to reproducing rather well the 0_{g.s.}^{+} and 1_{exc}^{+} binding energies, these four-body calculations demonstrate for the first time that the observed charge symmetry breaking splitting of mirror levels, reaching hundreds of keV for 0_{g.s.}^{+}, can be reproduced using realistic theoretical interaction models, although with a non-negligible momentum cutoff dependence. Our results are discussed in relation to recent measurements of the _{Λ}^{4}H(0_{g.s.}^{+}) binding energy at the Mainz Microtron [A. Esser et al. (A1 Collaboration), Phys. Rev. Lett. 114, 232501 (2015)] and the _{Λ}^{4}He(1_{exc}^{+}) excitation energy [T.O. Yamamoto et al. (J-PARC E13 Collaboration), Phys. Rev. Lett. 115, 222501 (2015)].