Neutron production driven by intense lasers utilizing inverse kinematic reactions is explored self-consistently by a combination of particle-in-cell simulations for laser-driven ion acceleration and Monte Carlo nuclear reaction simulations for neutron production. It is proposed that laser-driven light-sail acceleration from ultrathin lithium foils can provide an energetic lithium-ion beam as the projectile bombarding a light hydrocarbon target with sufficiently high flux for the inverse p(Li7,n) reaction to be efficiently achieved. Three-dimensional self-consistent simulations show that a forward-directed pulsed neutron source with ultrashort pulse duration 3 ns, small divergence angle 26°, and extremely high peak flux 3 × 1014n/(cm2⋅s) can be produced by petawatt lasers at intensities of 1021 W/cm2. These results indicate that a laser-driven neutron source based on inverse kinematics has promise as a novel compact pulsed neutron generator for practical applications, since the it can operate in a safe and repetitive way with almost no undesirable radiation.