We have performed the first direct measurement of two resonances of the ^{7}Be(α,γ)^{11}C reaction with unknown strengths using an intense radioactive ^{7}Be beam and the DRAGON recoil separator. We report on the first measurement of the 1155 and 1110 keV resonance strengths of 1.73±0.25(stat)±0.40(syst) eV and 125_{-25}^{+27}(stat)±15(syst) meV, respectively. The present results have reduced the uncertainty in the ^{7}Be(α,γ)^{11}C reaction rate to ∼9.4%-10.7% over T=1.5-3 GK, which is relevant for nucleosynthesis in the neutrino-driven outflows of core-collapse supernovae (νp process). We find no effect of the new, constrained reaction rate on νp-process nucleosynthesis.