Higher-order, non-Gaussian aspects of the large-scale structure carry valuable information on structure formation and cosmology, which is complementary to second-order statistics. In this work we measure second- and third-order weak-lensing aperture-mass moments from CFHTLenS and combine those with CMB anisotropy probes. The third moment is measured with a significance of $2σ$. The combined constraint on $Σ_8 = σ_8 (Ω_{\rm m}/0.27)^α$ is improved by 10%, in comparison to the second-order only, and the allowed ranges for $Ω_{\rm m}$ and $σ_8$ are substantially reduced. Including general triangles of the lensing bispectrum yields tighter constraints compared to probing mainly equilateral triangles. Second- and third-order CFHTLenS lensing measurements improve Planck CMB constraints on $Ω_{\rm m}$ and $σ_8$ by 26% for flat $Λ$CDM. For a model with free curvature, the joint CFHTLenS-Planck result is $Ω_{\rm m} = 0.28 ± 0.02$ (68% confidence), which is an improvement of 43% compared to Planck alone. We test how our results are potentially subject to three astrophysical sources of contamination: source-lens clustering, the intrinsic alignment of galaxy shapes, and baryonic effects. We explore future limitations of the cosmological use of third-order weak lensing, such as the nonlinear model and the Gaussianity of the likelihood function. ; Comment: Accepted by MNRAS. 21 pages, 14 figues, 8 tables. The data is available at http://www.cfhtlens.org . The software used for the cosmological analysis can be downloaded from http://cosmopmc.info