The HeII transverse proximity effect-enhanced HeII Ly{alpha} transmission in a background sightline caused by the ionizing radiation of a foreground quasar-offers a unique opportunity to probe the morphology of quasar-driven HeII reionization. We conduct a comprehensive spectroscopic survey to find z~3 quasars in the foreground of 22 background quasar sightlines with Hubble Space Telescope/COS HeII Ly{alpha} transmission spectra. With our two-tiered survey strategy, consisting of a deep pencil-beam survey and a shallow wide-field survey, we discover 131 new quasars, which we complement with known SDSS/BOSS quasars in our fields. Using a restricted sample of 66 foreground quasars with inferred HeII photoionization rates greater than the expected UV background at these redshifts ({Gamma}_QSO_^HeII^>5x10^-16^s^-1^) we perform the first statistical analysis of the HeII transverse proximity effect. Our results show qualitative evidence for a large object-to-object variance: among the four foreground quasars with the highest {Gamma}_QSO_^HeII^ only one (previously known) quasar is associated with a significant HeII transmission spike. We perform a stacking analysis to average down these fluctuations, and detect an excess in the average HeII transmission near the foreground quasars at 3{sigma} significance. This statistical evidence for the transverse proximity effect is corroborated by a clear dependence of the signal strength on {Gamma}_QSO_^HeII^. Our detection places a purely geometrical lower limit on the quasar lifetime of t_Q_>25Myr. Improved modeling would additionally constrain quasar obscuration and the mean free path of HeII-ionizing photons.