We present the simultaneous measurement of the interaction rates R pp , R Be , R pep of pp, 7 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 ðcpdÞ=100 ton as unit, we find R pp ¼ 134 AE 10ðstatÞ þ6 −10 ðsysÞ, R Be ¼ 48.3 AE 1.1ðstatÞ þ0.4 −0.7 ðsysÞ; and R HZ pep ¼ 2.43 AE 0.36ðstatÞ þ0.15 −0.22 ðsysÞ assuming the interaction rate R CNO of CNO-cycle (Carbon, Nitrogen, Oxigen) solar neutrinos according to the prediction of the high metallicity standard solar model, and R LZ pep ¼ 2.65 AE 0.36ðstatÞ þ0.15 −0.24 ðsysÞ according to that of the low metallicity model. An upper limit R CNO < 8.1 cpd=100 ton (95% C.L.) is obtained by setting in the fit a constraint on the ratio R pp =R pep (47.7 AE 0.8 cpd=100 ton or 47.5 AE 0.8 cpd=100 ton according to the high or low metallicity hypothesis).