We present an analysis of the exclusive B+→π+π-ℓ+νℓ decay, where ℓ represents an electron or a muon, with the assumption of charge-conjugation symmetry and lepton universality. The study exploits the full ϒ(4S) data sample collected by the Belle detector, corresponding to 711 fb-1 of integrated luminosity. Events are selected by fully reconstructing one B meson in hadronic decay modes, subsequently determining the properties of the other B meson. We extract the signal yields using a binned maximum-likelihood fit to the missing-mass squared distribution in bins of the invariant mass of the two pions or the momentum transfer squared. We measure a total branching fraction of B(B+→π+π-ℓ+νℓ)=[22.7-1.6+1.9(stat)±3.5(syst)]×10-5, where the uncertainties are statistical and systematic, respectively. This result is the first reported measurement of this decay.