We present a new measurement of the Hubble Constant H0 and other cosmological parameters based on the joint analysis of three multiply imaged quasar systems with measured gravitational time delays. First, we measure the time delay of HE 0435-1223 from 13-yr light curves obtained as part of the COSMOGRAIL project. Companion papers detail the modelling of the main deflectors and line-of-sight effects, and how these data are combined to determine the time-delay distance of HE 0435-1223. Crucially, the measurements are carried out blindly with respect to cosmological parameters in order to avoid confirmation bias. We then combine the time-delay distance of HE 0435-1223 with previous measurements from systems B1608+656 and RXJ1131-1231 to create a Time Delay Strong Lensing probe (TDSL). In flat Λ cold dark matter (ΛCDM) with free matter and energy density, we find H0 = 71.9-3.0+2.4 km s-1 Mpc-1 and ΩΛ = 0.62-0.35+0.24. This measurement is completely independent of, and in agreement with, the local distance ladder measurements of H0. We explore more general cosmological models combining TDSL with other probes, illustrating its power to break degeneracies inherent to other methods. The joint constraints from TDSL and Planck are H0 = 69.2-2.2+1.4 km s-1 Mpc-1, ΩΛ = 0.70-0.01+0.01 and Ωk = 0.003-0.006+0.004 in open ΛCDM and H0 = 79.0-4.2+4.4 km s-1 Mpc-1, Ωde = 0.77-0.03+0.02 and w = -1.38-0.16+0.14 in flat wCDM. In combination with Planck and baryon acoustic oscillation data, when relaxing the constraints on the numbers of relativistic species we find Neff = 3.34-0.21+0.21 in NeffΛCDM and when relaxing the total mass of neutrinos we find ∑mν =0.182 eV in mνλCDM. Finally, in an open wCDM in combination with Planck and cosmic microwave background lensing, we find H0 = 77.9-4.2+5.0 km s-1 Mpc-1, Ωde = 0.77-0.03+0.03, Ωk = -0.003-0.004+0.004 and w = -1.37-0.23+0.18.