The rate of neutrino-electron elastic scattering interactions from 862 keV Be-7 solar neutrinos in Borexino is determined to be 46.0 +/- 1.5(stat)(-1.6)(+1.5)(syst)counts/(day . 100 ton). This corresponds to a nu(e)-equivalent Be-7 solar neutrino flux of (3.10 +/- 0.15) x 10(9) cm(-2) s(-1) and, under the assumption of nu(e) transition to other active neutrino flavours, yields an electron neutrino survival probability of 0.51 +/- 0.07 at 862 keV. The no flavor change hypothesis is ruled out at 5.0 sigma. A global solar neutrino analysis with free fluxes determines Phi(pp) = 6.06(-0.66)(+0.02) x 10(10) cm(-2) s(-1) and Phi(CNO) < 1.3 x 10(9) cm(-2) s(-1) (95% C.L.). These results significantly improve the precision with which the Mikheyev-Smirnov-Wolfenstein large mixing angle neutrino oscillation model is experimentally tested at low energy.