We report a measurement of the neutrino-electron elastic scattering rate of 862 keV ⁷Be solar neutrinos based on a 165.4 kt d exposure of KamLAND. The observed rate is 592 ± 94 (kt d)⁻¹, which corresponds to an 862-keV ⁷Be solar neutrino flux of (3.26 ± 0.52) × 10⁹ cm⁻²s⁻¹, assuming a pure electron-flavor flux. Comparing this flux with the standard solar model prediction and further assuming three-flavor mixing, a vₑ survival probability of 0.66 ± 0.15 is determined from the KamLAND data. Utilizing a global three-flavor oscillation analysis, we obtain a total ⁷Be solar neutrino flux of (5.82 ± 1.02) × 10⁹ cm⁻²s⁻¹, which is consistent with the standard solar model predictions.