We report branching fraction measurements of the decays B+ -> eta l(+)v(l) and B+ -> eta'l(+)v(l) based on 711 fb(-1) of data collected near the (sic)(4S) resonance with the Belle experiment at the KEKB asymmetric-energy e(+)e(-) collider. This data sample contains 772 million B (B) over bar events. One of the two B mesons is fully reconstructed in a hadronic decay mode. Among the remaining ("signal-B") daughters, we search for the eta meson in two decay channels, eta -> gamma gamma and eta -> pi(+)pi(-)pi(0), and reconstruct the eta' meson in eta' -> eta pi(+)pi(-) with subsequent decay of the eta into gamma gamma Combining the two. modes and using an extended maximum likelihood, the B+ -> eta l(+)v(l) branching fraction is measured to be (4.2 +/- 1.1 (stat.) +/- 0.3 (syst.)) x 10(-5). For B+ -> eta l(+)v(l),we observe no significant signal and set an upper limit of 0.72 x 10(-4) at 90% confidence level.