We report measurements of the lifetimes of the Λc+, Ξc+ and Ξc0 charm baryons using proton-proton collision data at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3.0 fb-1, collected by the LHCb experiment. The charm baryons are reconstructed through the decays Λc+→pK-π+, Ξc+→pK-π+ and Ξc0→pK-K-π+, and originate from semimuonic decays of beauty baryons. The lifetimes are measured relative to that of the D+ meson, and are determined to be τΛc+=203.5±1.0±1.3±1.4 fs, τΞc+=456.8±3.5±2.9±3.1 fs, τΞc0=154.5±1.7±1.6±1.0 fs, where the uncertainties are statistical, systematic, and due to the uncertainty in the D+ lifetime. The measurements are approximately 3-4 times more precise than the current world average values. The Λc+ and Ξc+ lifetimes are in agreement with previous measurements; however, the Ξc0 baryon lifetime is approximately 3.3 standard deviations larger than the world average value.