We present the simultaneous measurement of the interaction rates ${R}_{pp}$, ${R}_{\mathrm{Be}}$, ${R}_{pep}$ of $pp$, $^{7}\mathrm{Be}$, and $pep$ solar neutrinos performed with a global fit to the Borexino data in an extended energy range (0.19--2.93) MeV with particular attention to details of the analysis methods. This result was obtained by analyzing 1291.51 days of Borexino Phase-II data, collected after an extensive scintillator purification campaign. Using counts per day $(\mathrm{cpd})/100\text{ }\mathrm{ton}$ as unit, we find ${R}_{pp}=134\ifmmode\pm\else\textpm\fi{}10(\mathrm{stat}{)}_{\ensuremath{-}10}^{+6}(\mathrm{sys})$, ${R}_{Be}=\phantom{\rule{0ex}{0ex}}48.3\ifmmode\pm\else\textpm\fi{}1.1(\mathrm{stat}{)}_{\ensuremath{-}0.7}^{+0.4}(\mathrm{sys})$; and ${R}_{pep}^{\mathrm{HZ}}=2.43\ifmmode\pm\else\textpm\fi{}0.36(\mathrm{stat}{)}_{\ensuremath{-}0.22}^{+0.15}(\mathrm{sys})$ assuming the interaction rate ${R}_{\mathrm{CNO}}$ of CNO-cycle (Carbon, Nitrogen, Oxigen) solar neutrinos according to the prediction of the high metallicity standard solar model, and ${R}_{pep}^{\mathrm{LZ}}=2.65\ifmmode\pm\else\textpm\fi{}0.36(\mathrm{stat}{)}_{\ensuremath{-}0.24}^{+0.15}(\mathrm{sys})$ according to that of the low metallicity model. An upper limit ${R}_{\mathrm{CNO}}<8.1\text{ }\mathrm{cpd}/100\text{ }\text{ }\mathrm{ton}$ (95% C.L.) is obtained by setting in the fit a constraint on the ratio ${R}_{pp}/{R}_{pep}$ ($47.7\ifmmode\pm\else\textpm\fi{}0.8\text{ }\text{ }\mathrm{cpd}/100\text{ }\mathrm{ton}$ or $47.5\ifmmode\pm\else\textpm\fi{}0.8\text{ }\text{ }\mathrm{cpd}/100\text{ }\text{ }\mathrm{ton}$ according to the high or low metallicity hypothesis).