In this paper, we report the first measurement of CNO solar neutrinos by Borexino that uses the correlated integrated directionality (CID) method, exploiting the subdominant Cherenkov light in the liquid scintillator detector. The directional information of the solar origin of the neutrinos is preserved by the early Cherenkov photons from the neutrino scattered electrons and is used to discriminate between signal and background. The directional information is independent from the spectral information on which the previous CNO solar neutrino measurements by Borexino were based, except for the selection of the energy region of interest. While the CNO spectral analysis could only be applied on the Phase-III dataset, the directional analysis can use the complete Borexino data taking period from 2007 to 2021. The absence of CNO neutrinos has been rejected with >5σ posterior probability using the Bayesian statistics. The directional CNO measurement is obtained without an external constraint on the Bi210 contamination of the liquid scintillator, which was applied in the spectral analysis approach. The final and the most precise CNO measurement of Borexino is then obtained by combining the new CID-based CNO result with an improved spectral fit of the Phase-III dataset. Including the statistical and the systematic errors, the extracted CNO interaction rate is R(CNO)=6.7-0.8+1.2 cpd/100 tonnes. Taking into account the neutrino flavor conversion, the resulting CNO neutrino flux at Earth is φCNO=6.7-0.8+1.2×108 cm-2 s-1, which is found to be in agreement with the high metallicity standard solar models. This outcome, combined with the Be7 and B8 fluxes measurements previously obtained by Borexino, can be used to disfavor the low metallicity SSM B16-AGSS09met model at 3.2σ CL, assuming the SSM B16-GS98 high metallicity model to be true. Also, the sum of C and N abundances in the solar core with respect to the H abundance is evaluated with improved precision, resulting in NCN=5.81-0.94+1.22×10-4, which is compatible with the high metallicity scenario and exhibits a 2σ tension with the low metallicity case. The results described in this work reinforce the role of directional Cherenkov light in large-scale liquid scintillator detectors and open up new avenues for the next-generation liquid scintillator or hybrid neutrino experiments. A particular relevance is expected for the latter detectors, which aim to combine the advantages from both Cherenkov-based and scintillation-based detection techniques.