We analyze the electrodynamic interaction of an incident terahertz electromagnetic wave with a current-carrying lateral double-quantum-wire superlattice (at normal incidence). The superlattice (in the x-y plane) is taken to consist of two parallel quantum-wire sublattices, each of period d, shifted with respect to each other by distance Δ in the transverse y direction. The parallel quantum wires of the sublattices are all oriented in the x direction. The two sublattices are taken to carry equal steady currents in opposite directions, and are coupled by Coulomb forces alone, with tunneling neglected. We recently showed that quasi-one-dimensional plasmons of such double-quantum-wire superlattice systems become unstable when the electron drift velocity falls between the phase velocities of the acoustic and optical plasmon modes of the Coulomb-coupled wire subsystems. Here, we employ a random phase approximation for plasmon dispersion taken jointly with the full system of Maxwell equations to describe the electrodynamic interaction of the incident terahertz electromagnetic radiation with the superlattice electron system. Coupling of the electromagnetic wave with the plasmon excitations is provided by introducing a metal grating with the grating stripes oriented perpendicular to the quantum wires. We have determined the transmission, absorption, and reflection coefficients for an incident terahertz electromagnetic wave propagating through the grating-superlattice system, demonstrating that amplification of the terahertz electromagnetic radiation occurs in the region of plasma instability. Our numerical calculations show that this effect occurs at experimentally achievable drift velocities in GaAs-based structures.