A first measurement of the longitudinal beam spin asymmetry A_{LU} in the semi-inclusive electroproduction of pairs of charged pions is reported. A_{LU} is a higher-twist observable and offers the cleanest access to the nucleon twist-3 parton distribution function e(x). Data have been collected in the Hall-B at Jefferson Lab by impinging a 5.498-GeV electron beam on a liquid-hydrogen target, and reconstructing the scattered electron and the pion pair with the CLAS detector. One-dimensional projections of the A_{LU}^{sinϕ_{R}} moments are extracted for the kinematic variables of interest in the valence quark region. The understanding of dihadron production is essential for the interpretation of observables in single-hadron production in semi-inclusive DIS, and pioneering measurements of single-spin asymmetries in dihadron production open a new avenue in studies of QCD dynamics.